RFE Dec 2025: Mesh patterns avoiding 321
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Sean A. Irvine
Dec 3, 2025, 4:26:01 PM12/3/25
to seq...@googlegroups.com
Hi,
This month I would like to draw attention to what is currently the largest group of OEIS sequences with identical names.
Provide more terms, distinct names, formulas, and/or programs for the 31 sequences based on Murray Tannock's MSc Thesis (available via the Links in the corresponding sequences). These sequences appear (along with many others) in Appendix B, starting on p. 47 of the thesis.
A289452, A289453, A289587, A289588, A289589, A289590, A289591, A289592, A289593, A289594, A289595, A289596, A289597, A289598, A289599, A289600, A289601, A289602, A289603, A289604, A289605, A289606, A289607, A289608, A289609, A289610, A289611, A289612, A289613, A289614, A289654.
The RFE from November also remains open, although M. H. Hasler (and others) have made some improvements to those sequences.
I track these requests for enhancement here:
https://oeis.org/wiki/User:Sean_A._Irvine/Requests_for_Enhancements#Requests_for_Enhancements
Sean.
This month I would like to draw attention to what is currently the largest group of OEIS sequences with identical names.
Provide more terms, distinct names, formulas, and/or programs for the 31 sequences based on Murray Tannock's MSc Thesis (available via the Links in the corresponding sequences). These sequences appear (along with many others) in Appendix B, starting on p. 47 of the thesis.
A289452, A289453, A289587, A289588, A289589, A289590, A289591, A289592, A289593, A289594, A289595, A289596, A289597, A289598, A289599, A289600, A289601, A289602, A289603, A289604, A289605, A289606, A289607, A289608, A289609, A289610, A289611, A289612, A289613, A289614, A289654.
The RFE from November also remains open, although M. H. Hasler (and others) have made some improvements to those sequences.
I track these requests for enhancement here:
https://oeis.org/wiki/User:Sean_A._Irvine/Requests_for_Enhancements#Requests_for_Enhancements
Sean.
Christian Sievers
Dec 4, 2025, 8:59:22 PM12/4/25
to SeqFan
Hello,
the name of these sequences should be something like
Number of permutations of length n that avoid the (classical) pattern 321
(or 231 in the case of the sequences from appendix B.1)
and the mesh pattern (tau,R).
(where tau = 12 or 21 and R is a subset of {0,1,2}^2 )
- except that often there is more than one mesh pattern that gives
the same terms. Worse, I'm not sure if the mesh patterns that give
the same first eleven terms are guaranteed to give the same full
sequence. Section 3.2 has subsections that begin with
"The following patterns are experimentally Wilf-equivalent up to
length 10 in Av(231) ..."
and then prove that they are indeed equivalent.
But that is only for the classical pattern 231, and indeed the
abstract only claims to "completely Wilf-classify mesh patterns
of length 2 when avoiding the classical pattern 231."
Except for the sequence 1,1,1,0,0,0,0,0,0,0,0, all sequences in
appendix B that have no "Related OEIS entry" in the table
(these are the new ones that we're talking about) correspond to
at most 8 "Number of patterns in class", as the column heading of
the table says. This is the number of mesh patterns that give the
particular sequence. In each table (for 231 and 321) their sum is
1024, corresponding to 2 options for tau and 2^9 options for R.
The offset of the sequences is given as 1 now, this should be 0.
The numbers correspond to lengths n=0..10, not n=1..11.
In Appendix C, the thesis explains that the python code used for it
can be found at github. The repository is still available and also
contains the TeX code of the thesis. I used it to sum the numbers
of patterns given in the tables.
I used my own code to check my understanding of the sequences.
I used clingo, which has recently become my "secret weapon"
which I used to compute a lot of my latest OEIS contributions,
often with embarrassingly trivial code.
(In Debian, you need the package gringo and its dependency clasp
to use clingo.)
I put the code at the end of this message.
It should be able to compute all the sequences of Appendix B.
You give it n, the classical pattern pi = 321 or 231, tau = 12 or 21,
and R encoded by a variable rc that has bit 3*I+J set to 1 if
(I,J) is in R, via a slightly clumsy option mechanism using
"-c <var>=<val>".
For example, section 3.2.1 gives six equivalent mesh patterns.
As the whole section 3.2, it uses the dominating pattern pi=231.
The position of the dots in the first mesh pattern show that it
has tau=12, and the set R has all possible elements except (1,1)
and is therefore encoded by rc=511-16.
If the program is in a file named mesh.lp, you can call clingo
like this to compute the term for n=10:
clingo -c n=10 -c pi=231 -c tau=12 -c rc=511-16 0 mesh.lp
You get a list of all the permutations that avoid pi and (tau,R)
and a summary telling that there were 15366 models, which is the
last number of the sequence at the end of the subsection.
The 0 tells clingo to enumerate all solutions, and if you don't
want to see them all, you can suppress the output of all answers
with the option -q.
For the third pattern in the same subsection, you would use tau=21
and rc=511-64.
Now one could systematically try all mesh pattern, or just play around.
Note that having the value for n=10 almost always determines the sequence.
Just playing around, it's not so easy to find a sequence that wasn't
already in the OEIS when the thesis was written.
But here is one I found: computing
clingo -c n=10 -c pi=321 -c tau=12 -c rc=234 -q 0 mesh.lp
gives 3740 models. Searching this number in the table in
When the computations become too slow, there are still options for
parallelization and solver tuning.
So it should be possible to find mesh patterns that give all these
sequences. But with so many sequences that already were in the OEIS,
one may assume that the remainig ones also have some independent
relevance.
All the best
Christian
------------------------------------------------------------
% p describes a permutation
{p(X,1..n)} = 1 :- X=1..n.
{p(1..n,Y)} = 1 :- Y=1..n.
% ... that avoids the classical pattern pi
:- pi=321, p(X1,Y1), p(X2,Y2), p(X3,Y3), X1<X2, X2<X3, Y3<Y2, Y2<Y1.
:- pi=231, p(X1,Y1), p(X2,Y2), p(X3,Y3), X1<X2, X2<X3, Y3<Y1, Y1<Y2.
% ... and also avoids the mesh pattern (tau,R)
% where R is given by the predicate r with
% r(I,J) <=> (I,J) \in R
:- p(X1,Y1), p(X2,Y2), X1<X2,
Y1<Y2 : tau=12;
Y2<Y1 : tau=21;
Z1 = #min{Z : Z=(Y1;Y2)},
Z2 = #max{Z : Z=(Y1;Y2)},
not r(I,J) : p(X,Y), o(X1,X2,X,I), o(Z1,Z2,Y,J).
% get the predicate r from its encoding rc via
% r(I,J) <=> bit 3*I+J in rc is 1
r(I,J) :- I=0..2, J=0..2, 2**(3*I+J)&rc>0.
% o(A,B,X,O) if 1 <= A < B <= n, X is a different element in 1..n, and
% o=0, 1 or 2 when X comes before, between, or after A and B, respectively.
o(A,B,X,0) :- X=1..n, A=X+1..n, B=A+1..n.
o(A,B,X,1) :- A=1..n, X=A+1..n, B=X+1..n.
o(A,B,X,2) :- A=1..n, B=A+1..n, X=B+1..n.
#show p/2.
------------------------------------------------------------
the name of these sequences should be something like
Number of permutations of length n that avoid the (classical) pattern 321
(or 231 in the case of the sequences from appendix B.1)
and the mesh pattern (tau,R).
(where tau = 12 or 21 and R is a subset of {0,1,2}^2 )
- except that often there is more than one mesh pattern that gives
the same terms. Worse, I'm not sure if the mesh patterns that give
the same first eleven terms are guaranteed to give the same full
sequence. Section 3.2 has subsections that begin with
"The following patterns are experimentally Wilf-equivalent up to
length 10 in Av(231) ..."
and then prove that they are indeed equivalent.
But that is only for the classical pattern 231, and indeed the
abstract only claims to "completely Wilf-classify mesh patterns
of length 2 when avoiding the classical pattern 231."
Except for the sequence 1,1,1,0,0,0,0,0,0,0,0, all sequences in
appendix B that have no "Related OEIS entry" in the table
(these are the new ones that we're talking about) correspond to
at most 8 "Number of patterns in class", as the column heading of
the table says. This is the number of mesh patterns that give the
particular sequence. In each table (for 231 and 321) their sum is
1024, corresponding to 2 options for tau and 2^9 options for R.
The offset of the sequences is given as 1 now, this should be 0.
The numbers correspond to lengths n=0..10, not n=1..11.
In Appendix C, the thesis explains that the python code used for it
can be found at github. The repository is still available and also
contains the TeX code of the thesis. I used it to sum the numbers
of patterns given in the tables.
I used my own code to check my understanding of the sequences.
I used clingo, which has recently become my "secret weapon"
which I used to compute a lot of my latest OEIS contributions,
often with embarrassingly trivial code.
(In Debian, you need the package gringo and its dependency clasp
to use clingo.)
I put the code at the end of this message.
It should be able to compute all the sequences of Appendix B.
You give it n, the classical pattern pi = 321 or 231, tau = 12 or 21,
and R encoded by a variable rc that has bit 3*I+J set to 1 if
(I,J) is in R, via a slightly clumsy option mechanism using
"-c <var>=<val>".
For example, section 3.2.1 gives six equivalent mesh patterns.
As the whole section 3.2, it uses the dominating pattern pi=231.
The position of the dots in the first mesh pattern show that it
has tau=12, and the set R has all possible elements except (1,1)
and is therefore encoded by rc=511-16.
If the program is in a file named mesh.lp, you can call clingo
like this to compute the term for n=10:
clingo -c n=10 -c pi=231 -c tau=12 -c rc=511-16 0 mesh.lp
You get a list of all the permutations that avoid pi and (tau,R)
and a summary telling that there were 15366 models, which is the
last number of the sequence at the end of the subsection.
The 0 tells clingo to enumerate all solutions, and if you don't
want to see them all, you can suppress the output of all answers
with the option -q.
For the third pattern in the same subsection, you would use tau=21
and rc=511-64.
Now one could systematically try all mesh pattern, or just play around.
Note that having the value for n=10 almost always determines the sequence.
Just playing around, it's not so easy to find a sequence that wasn't
already in the OEIS when the thesis was written.
But here is one I found: computing
clingo -c n=10 -c pi=321 -c tau=12 -c rc=234 -q 0 mesh.lp
gives 3740 models. Searching this number in the table in
B.2 gives one sequence (you can check smaller values) that
did not have a related OEIS entry. It is now A289587.
We can easily compute more terms:
a(11)=11602, a(12)=36357, a(13)=115049,...When the computations become too slow, there are still options for
parallelization and solver tuning.
So it should be possible to find mesh patterns that give all these
sequences. But with so many sequences that already were in the OEIS,
one may assume that the remainig ones also have some independent
relevance.
All the best
Christian
------------------------------------------------------------
% p describes a permutation
{p(X,1..n)} = 1 :- X=1..n.
{p(1..n,Y)} = 1 :- Y=1..n.
% ... that avoids the classical pattern pi
:- pi=321, p(X1,Y1), p(X2,Y2), p(X3,Y3), X1<X2, X2<X3, Y3<Y2, Y2<Y1.
:- pi=231, p(X1,Y1), p(X2,Y2), p(X3,Y3), X1<X2, X2<X3, Y3<Y1, Y1<Y2.
% ... and also avoids the mesh pattern (tau,R)
% where R is given by the predicate r with
% r(I,J) <=> (I,J) \in R
:- p(X1,Y1), p(X2,Y2), X1<X2,
Y1<Y2 : tau=12;
Y2<Y1 : tau=21;
Z1 = #min{Z : Z=(Y1;Y2)},
Z2 = #max{Z : Z=(Y1;Y2)},
not r(I,J) : p(X,Y), o(X1,X2,X,I), o(Z1,Z2,Y,J).
% get the predicate r from its encoding rc via
% r(I,J) <=> bit 3*I+J in rc is 1
r(I,J) :- I=0..2, J=0..2, 2**(3*I+J)&rc>0.
% o(A,B,X,O) if 1 <= A < B <= n, X is a different element in 1..n, and
% o=0, 1 or 2 when X comes before, between, or after A and B, respectively.
o(A,B,X,0) :- X=1..n, A=X+1..n, B=A+1..n.
o(A,B,X,1) :- A=1..n, X=A+1..n, B=X+1..n.
o(A,B,X,2) :- A=1..n, B=A+1..n, X=B+1..n.
#show p/2.
------------------------------------------------------------
Ruud H.G. van Tol
Dec 5, 2025, 6:45:45 AM12/5/25
to seq...@googlegroups.com
On 2025-12-03 22:25, Sean A. Irvine wrote:
> This month I would like to draw attention to what is currently the
> largest group of OEIS sequences with identical names.
>
> Provide more terms, distinct names, formulas, and/or programs for the
> 31 sequences based on Murray Tannock's MSc Thesis (available via the
> Links in the corresponding sequences). These sequences appear (along
> with many others) in Appendix B, starting on p. 47 of the thesis.
>
> A289452, A289453, A289587, A289588, A289589, A289590, A289591,
> A289592, A289593, A289594, A289595, A289596, A289597, A289598,
> A289599, A289600, A289601, A289602, A289603, A289604, A289605,
> A289606, A289607, A289608, A289609, A289610, A289611, A289612,
> A289613, A289614, A289654.
shows 6 more: A289446 - A289451
-- Ruud
Ruud H.G. van Tol
Dec 5, 2025, 10:27:34 AM12/5/25
to seq...@googlegroups.com
-- Ruud
Ruud H.G. van Tol
Dec 5, 2025, 10:32:15 AM12/5/25
to seq...@googlegroups.com
On 2025-12-05 12:45, 'Ruud H.G. van Tol' via SeqFan wrote:
treatment.
-- Ruud
Christian Sievers
Dec 5, 2025, 6:25:33 PM12/5/25
to SeqFan
So, I did the systematic computing thing (took less time than writing the last message),
and I can essentially reconstruct the two tables in appendix B. That is, I get the same
sequences and numbers of mesh patterns that create them. Of course I recorded which
pattern create which sequence. I'm giving this data for the second table (B.2) below.
The format is:
- one line of terms a(0)..a(10)
- one or two lines for tau=12 or 21, all the subsets R encoded as described in my previous
message that together with tau (and the dominating pattern pi=321) give this sequence,
and a count of this Rs, so: tau=...: R1 R2 .. Rn (#n)
(and you see the extra empty line).
Now we need a reasonable way to give this information in an OEIS entry.
I can provide the same information for the other table in B.1, with dominating pattern pi=231
(it is shorter).
Best
Christian
------------------------------------------------------------
1 1 1 0 0 0 0 0 0 0 0
tau=12: 0 1 2 3 8 9 16 17 18 19 24 25 32 33 34 35 40 41 48 49 50 51 56 57 128 129 130 131 136 137 144 145 146 147 152 153 256 257 258 259 264 265 272 274 280 288 289 290 291 296 297 304 306 312 384 385 386 387 392 393 400 402 408 (#63)
1 1 1 1 1 1 1 1 1 1 1
tau=12: 4 5 6 7 10 12 13 20 21 22 23 26 28 36 37 38 39 44 45 52 53 54 55 60 64 65 66 67 72 73 80 81 82 88 89 96 97 104 105 112 113 120 121 132 133 134 135 148 149 150 151 160 176 192 193 194 195 200 201 208 209 210 216 217 260 261 262 263 268 269 276 278 284 292 293 294 295 300 301 308 310 316 320 321 322 323 328 329 336 338 344 352 353 360 361 388 389 390 391 448 449 450 451 456 457 464 466 472 (#108)
tau=21: 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 36 40 44 48 52 56 60 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 100 104 108 112 116 120 124 128 130 132 134 144 146 148 150 160 164 176 180 192 194 196 198 208 210 212 214 224 228 240 244 (#72)
1 1 1 2 3 4 5 6 7 8 9
tau=12: 68 69 84 324 325 (#5)
1 1 1 1 2 3 6 11 22 44 90
tau=12: 42 58 138 154 162 168 178 184 (#8)
1 1 1 2 4 8 16 32 64 128 256
tau=12: 70 71 76 77 86 92 100 101 108 109 116 124 196 197 198 199 212 214 326 327 332 333 356 357 364 365 452 453 454 455 (#30)
1 1 1 1 3 6 13 28 60 129 277
tau=12: 98 140 (#2)
1 1 1 2 3 9 16 48 102 289 693
tau=12: 170 (#1)
1 1 1 2 4 9 21 51 127 323 835
tau=12: 14 30 46 62 74 90 164 166 180 182 186 202 218 224 232 240 248 (#17)
1 1 1 1 3 8 21 55 144 377 987
tau=12: 99 141 354 396 (#4)
1 1 1 1 3 7 19 53 153 453 1367
tau=12: 114 156 (#2)
1 1 1 1 2 5 14 42 132 429 1430
tau=12: 11 27 29 61 83 211 368 376 404 406 416 432 (#12)
1 1 1 1 3 10 30 84 227 603 1589
tau=12: 355 397 (#2)
1 1 1 2 5 13 34 89 233 610 1597
tau=12: 102 103 204 205 358 359 460 461 (#8)
1 1 1 2 3 7 19 56 174 561 1859
tau=12: 85 340 (#2)
1 1 1 2 5 13 36 103 303 910 2779
tau=12: 106 122 142 158 172 188 226 242 (#8)
1 1 1 1 3 9 28 90 297 1001 3432
tau=12: 43 59 139 155 418 424 434 440 (#8)
1 1 1 3 6 18 47 139 405 1225 3740
tau=12: 174 234 (#2)
1 1 1 3 7 19 53 153 453 1367 4191
tau=12: 190 250 (#2)
1 1 1 2 4 11 34 110 365 1234 4237
tau=12: 117 213 342 348 (#4)
1 1 1 1 3 10 33 111 379 1312 4596
tau=12: 161 266 (#2)
1 1 1 2 5 14 42 132 429 1430 4862
tau=12: 15 31 47 63 75 87 91 93 118 125 203 215 219 220 372 380 420 422 436 438 468 470 480 488 496 504 (#26)
tau=21: 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 34 37 38 41 45 49 50 53 54 57 61 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 98 101 102 105 109 113 114 117 118 121 125 129 131 133 135 136 140 145 147 149 151 152 156 193 195 197 199 200 204 209 211 213 215 216 220 256 258 260 262 264 268 272 274 276 278 280 284 288 292 296 300 304 308 312 316 320 322 324 326 328 332 336 338 340 342 344 348 352 356 360 364 368 372 376 380 384 386 388 390 400 402 404 406 416 420 432 436 448 450 452 454 464 466 468 470 480 484 496 500 (#144)
1 1 1 1 4 12 39 129 436 1498 5218
tau=12: 163 169 298 394 (#4)
1 1 1 2 5 15 48 159 538 1850 6446
tau=12: 165 167 225 233 270 302 330 458 (#8)
1 1 1 1 4 14 48 165 572 2002 7072
tau=12: 115 157 177 282 370 412 (#6)
1 1 1 2 6 18 57 186 622 2120 7338
tau=12: 173 227 362 398 (#4)
1 1 1 2 6 19 61 200 670 2286 7918
tau=21: 42 46 58 62 106 110 122 126 138 142 154 158 162 166 168 172 178 182 184 188 202 206 218 222 226 230 232 236 242 246 248 252 (#32)
1 1 1 1 5 17 57 193 662 2299 8073
tau=12: 179 185 314 410 (#4)
1 1 1 2 6 19 61 202 683 2349 8191
tau=12: 171 426 (#2)
1 1 1 2 6 19 62 207 704 2431 8502
tau=12: 107 123 143 159 428 444 482 498 (#8)
1 1 1 3 8 24 75 243 808 2742 9458
tau=12: 78 94 110 126 206 222 228 229 230 231 236 237 244 246 252 334 366 462 (#18)
1 1 1 2 7 22 71 235 794 2728 9503
tau=12: 187 442 (#2)
1 1 1 2 6 20 68 233 805 2807 9879
tau=12: 119 181 183 221 241 249 286 318 346 374 474 476 (#12)
1 1 1 3 8 25 80 264 890 3053 10622
tau=12: 175 235 430 490 (#4)
1 1 1 1 4 16 63 239 880 3184 11431
tau=12: 267 417 (#2)
1 1 1 2 8 26 85 283 959 3300 11505
tau=12: 189 243 378 414 (#4)
1 1 1 1 5 20 74 265 937 3304 11678
tau=12: 299 395 419 425 (#4)
1 1 1 2 6 21 75 266 938 3305 11679
tau=12: 271 303 331 421 423 459 481 489 (#8)
1 1 1 2 7 25 86 292 995 3425 11926
tau=12: 363 399 429 483 (#4)
1 1 1 3 9 28 90 297 1001 3432 11934
tau=12: 79 95 111 127 191 207 223 251 335 367 446 463 484 485 486 487 492 493 500 502 506 508 (#22)
tau=21: 35 39 43 47 51 55 59 63 99 103 107 111 115 119 123 127 137 139 141 143 153 155 157 159 201 203 205 207 217 219 221 223 290 294 306 310 354 358 370 374 392 396 408 412 418 422 424 428 434 438 440 444 456 460 472 476 482 486 488 492 498 502 504 508 (#64)
1 1 1 4 10 31 97 316 1054 3586 12394
tau=12: 238 (#1)
1 1 1 3 10 31 98 321 1078 3686 12789
tau=21: 170 174 186 190 234 238 250 254 (#8)
1 1 1 4 11 33 105 343 1148 3916 13563
tau=12: 254 (#1)
1 1 1 2 7 26 93 325 1129 3935 13813
tau=21: 161 165 177 181 225 229 241 245 266 270 282 286 330 334 346 350 (#16)
1 1 1 4 11 34 108 354 1187 4054 14054
tau=12: 239 494 (#2)
1 1 1 3 9 31 105 355 1210 4171 14543
tau=12: 245 350 (#2)
1 1 1 3 10 33 109 364 1233 4236 14740
tau=12: 247 253 382 478 (#4)
tau=21: 163 167 169 173 179 183 185 189 227 231 233 237 243 247 249 253 298 302 314 318 362 366 378 382 394 398 410 414 458 462 474 478 (#32)
1 1 1 2 9 33 113 381 1291 4425 15357
tau=12: 427 (#1)
1 1 1 3 10 34 114 382 1292 4426 15358
tau=12: 431 491 (#2)
1 1 1 4 12 37 118 387 1298 4433 15366
tau=12: 255 495 510 (#3)
tau=21: 171 175 187 191 235 239 251 255 426 430 442 446 490 494 506 510 (#16)
1 1 1 3 9 30 104 365 1286 4542 16092
tau=21: 257 261 273 277 321 325 337 341 (#8)
1 1 1 1 6 22 91 349 1277 4570 16235
tau=12: 273 (#1)
1 1 1 2 6 25 96 357 1289 4587 16258
tau=12: 277 337 (#2)
1 1 1 3 7 28 101 365 1301 4604 16281
tau=12: 341 (#1)
1 1 1 1 7 25 102 377 1339 4699 16496
tau=12: 275 281 305 401 (#4)
1 1 1 2 7 28 106 382 1345 4706 16504
tau=12: 279 309 345 465 (#4)
1 1 1 1 8 28 108 387 1354 4720 16524
tau=12: 313 403 (#2)
1 1 1 2 7 29 109 388 1355 4721 16525
tau=12: 285 339 369 405 (#4)
1 1 1 2 8 31 112 392 1360 4727 16532
tau=12: 317 377 407 467 (#4)
1 1 1 3 9 32 113 393 1361 4728 16533
tau=12: 343 349 373 469 (#4)
1 1 1 3 10 34 116 397 1366 4734 16540
tau=12: 381 471 (#2)
tau=21: 259 263 265 269 275 279 281 285 289 293 297 301 305 309 313 317 323 327 329 333 339 343 345 349 353 357 361 365 369 373 377 381 385 387 389 391 401 403 405 407 449 451 453 455 465 467 469 471 (#48)
1 1 1 1 8 31 116 407 1401 4825 16750
tau=12: 307 409 (#2)
1 1 1 2 8 32 117 408 1402 4826 16751
tau=12: 311 473 (#2)
1 1 1 2 9 34 122 417 1416 4846 16778
tau=12: 283 433 (#2)
1 1 1 2 10 37 126 422 1422 4853 16786
tau=12: 371 413 (#2)
1 1 1 3 10 37 126 422 1422 4853 16786
tau=12: 287 347 437 497 (#4)
1 1 1 2 11 37 126 422 1422 4853 16786
tau=12: 315 411 435 441 (#4)
1 1 1 3 11 38 127 423 1423 4854 16787
tau=12: 319 375 439 475 477 505 (#6)
tau=21: 267 271 283 287 331 335 347 351 417 421 433 437 481 485 497 501 (#16)
1 1 1 3 12 40 130 427 1428 4860 16794
tau=12: 379 415 445 499 (#4)
1 1 1 4 12 40 130 427 1428 4860 16794
tau=12: 351 501 (#2)
1 1 1 3 13 40 130 427 1428 4860 16794
tau=12: 443 (#1)
1 1 1 4 13 41 131 428 1429 4861 16795
tau=12: 383 447 479 503 507 509 (#6)
tau=21: 291 295 299 303 307 311 315 319 355 359 363 367 371 375 379 383 393 395 397 399 409 411 413 415 419 423 425 429 435 439 441 445 457 459 461 463 473 475 477 479 483 487 489 493 499 503 505 509 (#48)
1 1 1 5 14 42 132 429 1430 4862 16796
tau=12: 511 (#1)
tau=21: 427 431 443 447 491 495 507 511 (#8)
tau=12: 0 1 2 3 8 9 16 17 18 19 24 25 32 33 34 35 40 41 48 49 50 51 56 57 128 129 130 131 136 137 144 145 146 147 152 153 256 257 258 259 264 265 272 274 280 288 289 290 291 296 297 304 306 312 384 385 386 387 392 393 400 402 408 (#63)
1 1 1 1 1 1 1 1 1 1 1
tau=12: 4 5 6 7 10 12 13 20 21 22 23 26 28 36 37 38 39 44 45 52 53 54 55 60 64 65 66 67 72 73 80 81 82 88 89 96 97 104 105 112 113 120 121 132 133 134 135 148 149 150 151 160 176 192 193 194 195 200 201 208 209 210 216 217 260 261 262 263 268 269 276 278 284 292 293 294 295 300 301 308 310 316 320 321 322 323 328 329 336 338 344 352 353 360 361 388 389 390 391 448 449 450 451 456 457 464 466 472 (#108)
tau=21: 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 36 40 44 48 52 56 60 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 100 104 108 112 116 120 124 128 130 132 134 144 146 148 150 160 164 176 180 192 194 196 198 208 210 212 214 224 228 240 244 (#72)
1 1 1 2 3 4 5 6 7 8 9
tau=12: 68 69 84 324 325 (#5)
1 1 1 1 2 3 6 11 22 44 90
tau=12: 42 58 138 154 162 168 178 184 (#8)
1 1 1 2 4 8 16 32 64 128 256
tau=12: 70 71 76 77 86 92 100 101 108 109 116 124 196 197 198 199 212 214 326 327 332 333 356 357 364 365 452 453 454 455 (#30)
1 1 1 1 3 6 13 28 60 129 277
tau=12: 98 140 (#2)
1 1 1 2 3 9 16 48 102 289 693
tau=12: 170 (#1)
1 1 1 2 4 9 21 51 127 323 835
tau=12: 14 30 46 62 74 90 164 166 180 182 186 202 218 224 232 240 248 (#17)
1 1 1 1 3 8 21 55 144 377 987
tau=12: 99 141 354 396 (#4)
1 1 1 1 3 7 19 53 153 453 1367
tau=12: 114 156 (#2)
1 1 1 1 2 5 14 42 132 429 1430
tau=12: 11 27 29 61 83 211 368 376 404 406 416 432 (#12)
1 1 1 1 3 10 30 84 227 603 1589
tau=12: 355 397 (#2)
1 1 1 2 5 13 34 89 233 610 1597
tau=12: 102 103 204 205 358 359 460 461 (#8)
1 1 1 2 3 7 19 56 174 561 1859
tau=12: 85 340 (#2)
1 1 1 2 5 13 36 103 303 910 2779
tau=12: 106 122 142 158 172 188 226 242 (#8)
1 1 1 1 3 9 28 90 297 1001 3432
tau=12: 43 59 139 155 418 424 434 440 (#8)
1 1 1 3 6 18 47 139 405 1225 3740
tau=12: 174 234 (#2)
1 1 1 3 7 19 53 153 453 1367 4191
tau=12: 190 250 (#2)
1 1 1 2 4 11 34 110 365 1234 4237
tau=12: 117 213 342 348 (#4)
1 1 1 1 3 10 33 111 379 1312 4596
tau=12: 161 266 (#2)
1 1 1 2 5 14 42 132 429 1430 4862
tau=12: 15 31 47 63 75 87 91 93 118 125 203 215 219 220 372 380 420 422 436 438 468 470 480 488 496 504 (#26)
tau=21: 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 34 37 38 41 45 49 50 53 54 57 61 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 98 101 102 105 109 113 114 117 118 121 125 129 131 133 135 136 140 145 147 149 151 152 156 193 195 197 199 200 204 209 211 213 215 216 220 256 258 260 262 264 268 272 274 276 278 280 284 288 292 296 300 304 308 312 316 320 322 324 326 328 332 336 338 340 342 344 348 352 356 360 364 368 372 376 380 384 386 388 390 400 402 404 406 416 420 432 436 448 450 452 454 464 466 468 470 480 484 496 500 (#144)
1 1 1 1 4 12 39 129 436 1498 5218
tau=12: 163 169 298 394 (#4)
1 1 1 2 5 15 48 159 538 1850 6446
tau=12: 165 167 225 233 270 302 330 458 (#8)
1 1 1 1 4 14 48 165 572 2002 7072
tau=12: 115 157 177 282 370 412 (#6)
1 1 1 2 6 18 57 186 622 2120 7338
tau=12: 173 227 362 398 (#4)
1 1 1 2 6 19 61 200 670 2286 7918
tau=21: 42 46 58 62 106 110 122 126 138 142 154 158 162 166 168 172 178 182 184 188 202 206 218 222 226 230 232 236 242 246 248 252 (#32)
1 1 1 1 5 17 57 193 662 2299 8073
tau=12: 179 185 314 410 (#4)
1 1 1 2 6 19 61 202 683 2349 8191
tau=12: 171 426 (#2)
1 1 1 2 6 19 62 207 704 2431 8502
tau=12: 107 123 143 159 428 444 482 498 (#8)
1 1 1 3 8 24 75 243 808 2742 9458
tau=12: 78 94 110 126 206 222 228 229 230 231 236 237 244 246 252 334 366 462 (#18)
1 1 1 2 7 22 71 235 794 2728 9503
tau=12: 187 442 (#2)
1 1 1 2 6 20 68 233 805 2807 9879
tau=12: 119 181 183 221 241 249 286 318 346 374 474 476 (#12)
1 1 1 3 8 25 80 264 890 3053 10622
tau=12: 175 235 430 490 (#4)
1 1 1 1 4 16 63 239 880 3184 11431
tau=12: 267 417 (#2)
1 1 1 2 8 26 85 283 959 3300 11505
tau=12: 189 243 378 414 (#4)
1 1 1 1 5 20 74 265 937 3304 11678
tau=12: 299 395 419 425 (#4)
1 1 1 2 6 21 75 266 938 3305 11679
tau=12: 271 303 331 421 423 459 481 489 (#8)
1 1 1 2 7 25 86 292 995 3425 11926
tau=12: 363 399 429 483 (#4)
1 1 1 3 9 28 90 297 1001 3432 11934
tau=12: 79 95 111 127 191 207 223 251 335 367 446 463 484 485 486 487 492 493 500 502 506 508 (#22)
tau=21: 35 39 43 47 51 55 59 63 99 103 107 111 115 119 123 127 137 139 141 143 153 155 157 159 201 203 205 207 217 219 221 223 290 294 306 310 354 358 370 374 392 396 408 412 418 422 424 428 434 438 440 444 456 460 472 476 482 486 488 492 498 502 504 508 (#64)
1 1 1 4 10 31 97 316 1054 3586 12394
tau=12: 238 (#1)
1 1 1 3 10 31 98 321 1078 3686 12789
tau=21: 170 174 186 190 234 238 250 254 (#8)
1 1 1 4 11 33 105 343 1148 3916 13563
tau=12: 254 (#1)
1 1 1 2 7 26 93 325 1129 3935 13813
tau=21: 161 165 177 181 225 229 241 245 266 270 282 286 330 334 346 350 (#16)
1 1 1 4 11 34 108 354 1187 4054 14054
tau=12: 239 494 (#2)
1 1 1 3 9 31 105 355 1210 4171 14543
tau=12: 245 350 (#2)
1 1 1 3 10 33 109 364 1233 4236 14740
tau=12: 247 253 382 478 (#4)
tau=21: 163 167 169 173 179 183 185 189 227 231 233 237 243 247 249 253 298 302 314 318 362 366 378 382 394 398 410 414 458 462 474 478 (#32)
1 1 1 2 9 33 113 381 1291 4425 15357
tau=12: 427 (#1)
1 1 1 3 10 34 114 382 1292 4426 15358
tau=12: 431 491 (#2)
1 1 1 4 12 37 118 387 1298 4433 15366
tau=12: 255 495 510 (#3)
tau=21: 171 175 187 191 235 239 251 255 426 430 442 446 490 494 506 510 (#16)
1 1 1 3 9 30 104 365 1286 4542 16092
tau=21: 257 261 273 277 321 325 337 341 (#8)
1 1 1 1 6 22 91 349 1277 4570 16235
tau=12: 273 (#1)
1 1 1 2 6 25 96 357 1289 4587 16258
tau=12: 277 337 (#2)
1 1 1 3 7 28 101 365 1301 4604 16281
tau=12: 341 (#1)
1 1 1 1 7 25 102 377 1339 4699 16496
tau=12: 275 281 305 401 (#4)
1 1 1 2 7 28 106 382 1345 4706 16504
tau=12: 279 309 345 465 (#4)
1 1 1 1 8 28 108 387 1354 4720 16524
tau=12: 313 403 (#2)
1 1 1 2 7 29 109 388 1355 4721 16525
tau=12: 285 339 369 405 (#4)
1 1 1 2 8 31 112 392 1360 4727 16532
tau=12: 317 377 407 467 (#4)
1 1 1 3 9 32 113 393 1361 4728 16533
tau=12: 343 349 373 469 (#4)
1 1 1 3 10 34 116 397 1366 4734 16540
tau=12: 381 471 (#2)
tau=21: 259 263 265 269 275 279 281 285 289 293 297 301 305 309 313 317 323 327 329 333 339 343 345 349 353 357 361 365 369 373 377 381 385 387 389 391 401 403 405 407 449 451 453 455 465 467 469 471 (#48)
1 1 1 1 8 31 116 407 1401 4825 16750
tau=12: 307 409 (#2)
1 1 1 2 8 32 117 408 1402 4826 16751
tau=12: 311 473 (#2)
1 1 1 2 9 34 122 417 1416 4846 16778
tau=12: 283 433 (#2)
1 1 1 2 10 37 126 422 1422 4853 16786
tau=12: 371 413 (#2)
1 1 1 3 10 37 126 422 1422 4853 16786
tau=12: 287 347 437 497 (#4)
1 1 1 2 11 37 126 422 1422 4853 16786
tau=12: 315 411 435 441 (#4)
1 1 1 3 11 38 127 423 1423 4854 16787
tau=12: 319 375 439 475 477 505 (#6)
tau=21: 267 271 283 287 331 335 347 351 417 421 433 437 481 485 497 501 (#16)
1 1 1 3 12 40 130 427 1428 4860 16794
tau=12: 379 415 445 499 (#4)
1 1 1 4 12 40 130 427 1428 4860 16794
tau=12: 351 501 (#2)
1 1 1 3 13 40 130 427 1428 4860 16794
tau=12: 443 (#1)
1 1 1 4 13 41 131 428 1429 4861 16795
tau=12: 383 447 479 503 507 509 (#6)
tau=21: 291 295 299 303 307 311 315 319 355 359 363 367 371 375 379 383 393 395 397 399 409 411 413 415 419 423 425 429 435 439 441 445 457 459 461 463 473 475 477 479 483 487 489 493 499 503 505 509 (#48)
1 1 1 5 14 42 132 429 1430 4862 16796
tau=12: 511 (#1)
tau=21: 427 431 443 447 491 495 507 511 (#8)
Christian Sievers
Dec 7, 2025, 5:51:24 PM12/7/25
to SeqFan
Yes, A289612 is indeed an interessting case. It appears in appendix B.2 with the expected three initial 1s and the
additional value 16540. It has an entry in the "Related OEIS entry" column of the table: A000079.
That sequence also appears in this table for the sequence 1,1,1,2,4,8,16,32,64,128,256 which makes sense,
because it is powrs of two. In its history, you see that A289612 had the name "Not A79" temporarily.
However, I'm sure giving that sequence as *related* was intentional. As is noted in the comments, the terms
(with the offset such that a(1)=3) match C(n+2)-2^n, where C(n) is the Catalan number.
(This is also true for the omitted term and the next four terms.)
Permutations that avoid only the pattern 321 (or 231) are counted by the Catalan numbers, so it is not surprising
that some sequences counting permutations that avoid 321 and something else are best described as
"Catalan numbers minus some reasonable sequence".
It makes sense to call the counting sequence *related* to this other sequence.
This happened more than once. A000027 (the positive integers) has been given as related OEIS entry for the
sequence 1,1,1,2,3,4,5,6,7,8,9, but also to 1,1,1,3,11,38,127,423,1423,4854,16787. This sequence does not only
appear in B.2, but also in B.1 and in subsection 3.2.4, where it is identified as C_k-(k-1) and as
C_n - A000027 offset 2.
(This sequence has its own entry, A273526, with two leading 1s and offset 1. If the other sequences are fixed
to have offset 0, as I have argued for, I think this one should have a(0)=1 added to it.)
So if another "related OEIS entry" doesn't seem to fit, the explanation may be that the sequence is the Catalan
numbers minus this (possibly shifted) related sequence.
BTW, I computed the next four terms for all the sequences related to 321, and the next three terms for 231,
and for all the mesh patterns that originally gave the same sequence, the new terms were also the same.
All the best
Christian
Ruud H.G. van Tol
Dec 7, 2025, 6:31:50 PM12/7/25
to seq...@googlegroups.com
On 2025-12-07 23:51, Christian Sievers wrote:
> [...]
> Permutations that avoid only the pattern 321 (or 231) are counted by
> the Catalan numbers, so it is not surprising
> that some sequences counting permutations that avoid 321 and something
> else are best described as
> "Catalan numbers minus some reasonable sequence". [...]
> the Catalan numbers, so it is not surprising
> that some sequences counting permutations that avoid 321 and something
> else are best described as
For A289598
1, 1, 1, 2, 7, 25, 86, 292, 995, 3425, 11926, ...
I added a Formula:
Conjecture: For n >= 2, a(n) = C(n-1) - C(n-2) - n + 3, where C = A000108.
-- Ruud
Christian Sievers
Dec 7, 2025, 7:35:12 PM12/7/25
to SeqFan
> For A289598
> 1, 1, 1, 2, 7, 25, 86, 292, 995, 3425, 11926, ...
> I added a Formula:
>
> Conjecture: For n >= 2, a(n) = C(n-1) - C(n-2) - n + 3, where C = A000108.
> 1, 1, 1, 2, 7, 25, 86, 292, 995, 3425, 11926, ...
> I added a Formula:
>
> Conjecture: For n >= 2, a(n) = C(n-1) - C(n-2) - n + 3, where C = A000108.
That agrees with my further terms (when using the current offset).
I guess it's useful if I just give the extended sequences.
I have them in the same order as in the table in B.2, that is
first sorted by a(10), then a(9), and so on.
(Does anyone understand the order of the table in B.1?)
Here they are:
1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 2 3 4 5 6 7 8 9 10 11 12 13
1 1 1 1 2 3 6 11 22 44 90 187 392 832 1778
1 1 1 2 4 8 16 32 64 128 256 512 1024 2048 4096
1 1 1 1 3 6 13 28 60 129 277 595 1278 2745 5896
1 1 1 2 3 9 16 48 102 289 693 1902 4883 13296 35415
1 1 1 2 4 9 21 51 127 323 835 2188 5798 15511 41835
1 1 1 1 3 8 21 55 144 377 987 2584 6765 17711 46368
1 1 1 1 3 7 19 53 153 453 1367 4191 13015 40857 129441
1 1 1 1 2 5 14 42 132 429 1430 4862 16796 58786 208012
1 1 1 1 3 10 30 84 227 603 1589 4172 10936 28646 75013
1 1 1 2 5 13 34 89 233 610 1597 4181 10946 28657 75025
1 1 1 2 3 7 19 56 174 561 1859 6292 21658 75582 266798
1 1 1 2 5 13 36 103 303 910 2779 8603 26936 85149 271389
1 1 1 1 3 9 28 90 297 1001 3432 11934 41990 149226 534888
1 1 1 3 6 18 47 139 405 1225 3740 11602 36357 115049 366969
1 1 1 3 7 19 53 153 453 1367 4191 13015 40857 129441 413337
1 1 1 2 4 11 34 110 365 1234 4237 14741 51869 184299 660401
1 1 1 1 3 10 33 111 379 1312 4596 16266 58082 209010 757259
1 1 1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900
1 1 1 1 4 12 39 129 436 1498 5218 18386 65420 234734 848403
1 1 1 2 5 15 48 159 538 1850 6446 22712 80794 289804 1047063
1 1 1 1 4 14 48 165 572 2002 7072 25194 90440 326876 1188640
1 1 1 2 6 18 57 186 622 2120 7338 25724 91144 325878 1174281
1 1 1 2 6 19 61 200 670 2286 7918 27770 98424 351983 1268541
1 1 1 1 5 17 57 193 662 2299 8073 28626 102374 368866 1337866
1 1 1 2 6 19 61 202 683 2349 8191 28892 102904 369570 1336868
1 1 1 2 6 19 62 207 704 2431 8502 30056 107236 385662 1396652
1 1 1 3 8 24 75 243 808 2742 9458 33062 116868 417022 1500159
1 1 1 2 7 22 71 235 794 2728 9503 33488 119170 427652 1545878
1 1 1 2 6 20 68 233 805 2807 9879 35073 125513 452389 1641029
1 1 1 3 8 25 80 264 890 3053 10622 37394 132960 476806 1722530
1 1 1 1 4 16 63 239 880 3184 11431 40976 147189 530804 1923361
1 1 1 2 8 26 85 283 959 3300 11505 40560 144364 518092 1872754
1 1 1 1 5 20 74 265 937 3304 11678 41478 148202 532840 1927444
1 1 1 2 6 21 75 266 938 3305 11679 41479 148203 532841 1927445
1 1 1 2 7 25 86 292 995 3425 11926 41981 149216 534877 1931528
1 1 1 3 9 28 90 297 1001 3432 11934 41990 149226 534888 1931540
1 1 1 4 10 31 97 316 1054 3586 12394 43396 153604 548731 1975873
1 1 1 3 10 31 98 321 1078 3686 12789 44919 159407 570704 2058817
1 1 1 4 11 33 105 343 1148 3916 13563 47571 168625 603130 2174041
1 1 1 2 7 26 93 325 1129 3935 13813 48885 174397 626785 2267813
1 1 1 4 11 34 108 354 1187 4054 14054 49328 174950 626032 2257418
1 1 1 3 9 31 105 355 1210 4171 14543 51242 182242 653482 2360214
1 1 1 3 10 33 109 364 1233 4236 14740 51868 184298 660400 2383928
1 1 1 2 9 33 113 381 1291 4425 15357 53914 191205 684102 2466415
1 1 1 3 10 34 114 382 1292 4426 15358 53915 191206 684103 2466416
1 1 1 4 12 37 118 387 1298 4433 15366 53924 191216 684114 2466428
1 1 1 3 9 30 104 365 1286 4542 16092 57250 204684 735732 2659080
1 1 1 1 6 22 91 349 1277 4570 16235 57698 205883 738704 2666127
1 1 1 2 6 25 96 357 1289 4587 16258 57728 205921 738751 2666184
1 1 1 3 7 28 101 365 1301 4604 16281 57758 205959 738798 2666241
1 1 1 1 7 25 102 377 1339 4699 16496 58220 206923 740775 2670254
1 1 1 2 7 28 106 382 1345 4706 16504 58229 206933 740786 2670266
1 1 1 1 8 28 108 387 1354 4720 16524 58256 206968 740830 2670320
1 1 1 2 7 29 109 388 1355 4721 16525 58257 206969 740831 2670321
1 1 1 2 8 31 112 392 1360 4727 16532 58265 206978 740841 2670332
1 1 1 3 9 32 113 393 1361 4728 16533 58266 206979 740842 2670333
1 1 1 3 10 34 116 397 1366 4734 16540 58274 206988 740852 2670344
1 1 1 1 8 31 116 407 1401 4825 16750 58730 207945 742821 2674348
1 1 1 2 8 32 117 408 1402 4826 16751 58731 207946 742822 2674349
1 1 1 2 9 34 122 417 1416 4846 16778 58766 207990 742876 2674414
1 1 1 2 10 37 126 422 1422 4853 16786 58775 208000 742887 2674426
1 1 1 3 10 37 126 422 1422 4853 16786 58775 208000 742887 2674426
1 1 1 2 11 37 126 422 1422 4853 16786 58775 208000 742887 2674426
1 1 1 3 11 38 127 423 1423 4854 16787 58776 208001 742888 2674427
1 1 1 3 12 40 130 427 1428 4860 16794 58784 208010 742898 2674438
1 1 1 4 12 40 130 427 1428 4860 16794 58784 208010 742898 2674438
1 1 1 3 13 40 130 427 1428 4860 16794 58784 208010 742898 2674438
1 1 1 4 13 41 131 428 1429 4861 16795 58785 208011 742899 2674439
1 1 1 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 2 3 4 5 6 7 8 9 10 11 12 13
1 1 1 1 2 3 6 11 22 44 90 187 392 832 1778
1 1 1 2 4 8 16 32 64 128 256 512 1024 2048 4096
1 1 1 1 3 6 13 28 60 129 277 595 1278 2745 5896
1 1 1 2 3 9 16 48 102 289 693 1902 4883 13296 35415
1 1 1 2 4 9 21 51 127 323 835 2188 5798 15511 41835
1 1 1 1 3 8 21 55 144 377 987 2584 6765 17711 46368
1 1 1 1 3 7 19 53 153 453 1367 4191 13015 40857 129441
1 1 1 1 2 5 14 42 132 429 1430 4862 16796 58786 208012
1 1 1 1 3 10 30 84 227 603 1589 4172 10936 28646 75013
1 1 1 2 5 13 34 89 233 610 1597 4181 10946 28657 75025
1 1 1 2 3 7 19 56 174 561 1859 6292 21658 75582 266798
1 1 1 2 5 13 36 103 303 910 2779 8603 26936 85149 271389
1 1 1 1 3 9 28 90 297 1001 3432 11934 41990 149226 534888
1 1 1 3 6 18 47 139 405 1225 3740 11602 36357 115049 366969
1 1 1 3 7 19 53 153 453 1367 4191 13015 40857 129441 413337
1 1 1 2 4 11 34 110 365 1234 4237 14741 51869 184299 660401
1 1 1 1 3 10 33 111 379 1312 4596 16266 58082 209010 757259
1 1 1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900
1 1 1 1 4 12 39 129 436 1498 5218 18386 65420 234734 848403
1 1 1 2 5 15 48 159 538 1850 6446 22712 80794 289804 1047063
1 1 1 1 4 14 48 165 572 2002 7072 25194 90440 326876 1188640
1 1 1 2 6 18 57 186 622 2120 7338 25724 91144 325878 1174281
1 1 1 2 6 19 61 200 670 2286 7918 27770 98424 351983 1268541
1 1 1 1 5 17 57 193 662 2299 8073 28626 102374 368866 1337866
1 1 1 2 6 19 61 202 683 2349 8191 28892 102904 369570 1336868
1 1 1 2 6 19 62 207 704 2431 8502 30056 107236 385662 1396652
1 1 1 3 8 24 75 243 808 2742 9458 33062 116868 417022 1500159
1 1 1 2 7 22 71 235 794 2728 9503 33488 119170 427652 1545878
1 1 1 2 6 20 68 233 805 2807 9879 35073 125513 452389 1641029
1 1 1 3 8 25 80 264 890 3053 10622 37394 132960 476806 1722530
1 1 1 1 4 16 63 239 880 3184 11431 40976 147189 530804 1923361
1 1 1 2 8 26 85 283 959 3300 11505 40560 144364 518092 1872754
1 1 1 1 5 20 74 265 937 3304 11678 41478 148202 532840 1927444
1 1 1 2 6 21 75 266 938 3305 11679 41479 148203 532841 1927445
1 1 1 2 7 25 86 292 995 3425 11926 41981 149216 534877 1931528
1 1 1 3 9 28 90 297 1001 3432 11934 41990 149226 534888 1931540
1 1 1 4 10 31 97 316 1054 3586 12394 43396 153604 548731 1975873
1 1 1 3 10 31 98 321 1078 3686 12789 44919 159407 570704 2058817
1 1 1 4 11 33 105 343 1148 3916 13563 47571 168625 603130 2174041
1 1 1 2 7 26 93 325 1129 3935 13813 48885 174397 626785 2267813
1 1 1 4 11 34 108 354 1187 4054 14054 49328 174950 626032 2257418
1 1 1 3 9 31 105 355 1210 4171 14543 51242 182242 653482 2360214
1 1 1 3 10 33 109 364 1233 4236 14740 51868 184298 660400 2383928
1 1 1 2 9 33 113 381 1291 4425 15357 53914 191205 684102 2466415
1 1 1 3 10 34 114 382 1292 4426 15358 53915 191206 684103 2466416
1 1 1 4 12 37 118 387 1298 4433 15366 53924 191216 684114 2466428
1 1 1 3 9 30 104 365 1286 4542 16092 57250 204684 735732 2659080
1 1 1 1 6 22 91 349 1277 4570 16235 57698 205883 738704 2666127
1 1 1 2 6 25 96 357 1289 4587 16258 57728 205921 738751 2666184
1 1 1 3 7 28 101 365 1301 4604 16281 57758 205959 738798 2666241
1 1 1 1 7 25 102 377 1339 4699 16496 58220 206923 740775 2670254
1 1 1 2 7 28 106 382 1345 4706 16504 58229 206933 740786 2670266
1 1 1 1 8 28 108 387 1354 4720 16524 58256 206968 740830 2670320
1 1 1 2 7 29 109 388 1355 4721 16525 58257 206969 740831 2670321
1 1 1 2 8 31 112 392 1360 4727 16532 58265 206978 740841 2670332
1 1 1 3 9 32 113 393 1361 4728 16533 58266 206979 740842 2670333
1 1 1 3 10 34 116 397 1366 4734 16540 58274 206988 740852 2670344
1 1 1 1 8 31 116 407 1401 4825 16750 58730 207945 742821 2674348
1 1 1 2 8 32 117 408 1402 4826 16751 58731 207946 742822 2674349
1 1 1 2 9 34 122 417 1416 4846 16778 58766 207990 742876 2674414
1 1 1 2 10 37 126 422 1422 4853 16786 58775 208000 742887 2674426
1 1 1 3 10 37 126 422 1422 4853 16786 58775 208000 742887 2674426
1 1 1 2 11 37 126 422 1422 4853 16786 58775 208000 742887 2674426
1 1 1 3 11 38 127 423 1423 4854 16787 58776 208001 742888 2674427
1 1 1 3 12 40 130 427 1428 4860 16794 58784 208010 742898 2674438
1 1 1 4 12 40 130 427 1428 4860 16794 58784 208010 742898 2674438
1 1 1 3 13 40 130 427 1428 4860 16794 58784 208010 742898 2674438
1 1 1 4 13 41 131 428 1429 4861 16795 58785 208011 742899 2674439
1 1 1 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440
Bye
Christian
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