amazing long cycle of Collatz-type function based on Lucas representation

[Geoffrey Caveney]

Sequence A130310 describes the unique "minimal (or "greedy") Lucas representation of n, in which L(0) = 2 and L(2) = 3 are not allowed in the same representation (hence the correct representation of the integer 5 is 1010 rather than 101). A binary system of integers with Lucas numbers (A000032) as a base."

This is analogous to the Zeckendorf representation of integers as sums of non-consecutive Fibonacci numbers, but with Lucas numbers, the additional restriction prohibiting the co-occurrence of 2 and 3 is necessary to make the Lucas representation unique, as shown by Brown in 1969. Relevant information and sources are available at A342089, the sequence of numbers that have two such Lucas representations without the restriction on 2 and 3.

Define a number as "Lucas-even" if the last digit of its Lucas representation is 0 AND the removal of the last digit 0 produces a valid minimal Lucas representation of a smaller integer. The second condition is necessary because removing the final 0 from representations ending in ...1010 is NOT valid -- the resulting ...101 would be equivalent to the co-occurrence of 3 and 2 and thus it is not allowed.

Define a number as "Lucas-odd" either if the last digit of its Lucas representation is 1 OR if the last four digits of its Lucas representation are 1010.

Now define a Collatz-type function based on this Lucas representation:

If n is Lucas-even, remove the last digit 0 of its Lucas representation to obtain f(n) ;

If n is Lucas-odd, f(n) = 2n.

Note that removing the last digit 0 is approximately equivalent to division by phi = 1.618....

As in the classical Collatz function, the orbit of an integer is the sequence produced by the repeated iteration of the function.

Two simple cycles of this function are 2, 4, 3, 1, 2, ... and 5, 10, 5, .... Many small integers have orbits reaching one of these two cycles, such as 36, 22, 14, 8, 6, 12, 9, 18, 11, 7, 4, 3, 1, 2, ...;  28, 16, 32, 19, 13, 26, 17, 34, 68, 41, 27, 54, 33, 21, 12, 9, 18, 11, 7, 4, 3, 1, 2, ...;  and 30, 20, 40, 25, 15, 10, 5, ....

But there is also an amazingly long cycle beginning with 23, which is the reason for this post:

23, 46, 92, 184, 113, 71, 142, 89, 178, 111, 222, 444, 888, 1776, 1097, 679, 1358, 2716, 5432, 10864, 6713, 13426, 8297, 5129, 10258, 6341, 12682, 7839, 15678, 9691, 19382, 11979, 7403, 4576, 2827, 5654, 3493, 6986, 4317, 2669, 5338, 10676, 6597, 13194, 8153, 16306, 10077, 6229, 12458, 7701, 15402, 9519, 5883, 3636, 2247, 1389, 858, 531, 327, 654, 403, 806, 497, 994, 613, 1226, 757, 469, 938, 581, 1162, 719, 443, 886, 547, 339, 678, 1356, 837, 519, 1038, 641, 397, 244, 488, 301, 187, 374, 748, 461, 922, 569, 353, 706, 437, 269, 538, 1076, 2152, 1329, 823, 1646, 1017, 629, 388, 241, 482, 299, 598, 371, 742, 459, 283, 176, 352, 219, 438, 271, 167, 104, 63, 126, 77, 49, 98, 61, 37, 24, 48, 31, 62, 39, 23, ...

This cycle comprises 132 integers, including 16 integers < 100, 65 integers between 100 and 1,000, 38 integers between 1,000 and 10,000, and 13 integers > 10,000. The largest integer in this cycle is 19,382, the 30th iterate of 23. 

An interesting value in the cycle is 15,402 = 15,127 + 199 + 76, which gives the Lucas representation 100000000101000000000. This is the first of 8 consecutive Lucas-even numbers in the sequence, resulting in the much smaller Lucas-odd number 327 = 322 + 4 + 1, or 1000000001010 in Lucas representation.

For the record, believe it or not, I did NOT use any AI program to discover or compute this cycle. I was simply exploring the function out of curiosity and testing the orbits of small integers when I stumbled across the orbit of 23. It was actually not too difficult to compute each term by utilizing the b-file of A130310 for integers up to 10,000.

Geoffrey Caveney

[Geoffrey Caveney]

Incredibly, this Lucas-Collatz function has another very long cycle beginning with 70. This cycle contains 122 integers and reaches several values greater than 1 million, the largest of which is 1,863,240.

70, 140, 280, 560, 345, 690, 1380, 2760, 1705, 1055, 2110, 1305, 2610, 5220, 3225, 6450, 3985, 7970, 4925, 3045, 6090, 3765, 7530, 4655, 9310, 5755, 11510, 7115, 14230, 8795, 5435, 3360, 6720, 13440, 8305, 16610, 10265, 6345, 12690, 25380, 15685, 9695, 19390, 11985, 23970, 14815, 9155, 18310, 11315, 22630, 13985, 27970, 55940, 111880, 69145, 42735, 85470, 52825, 105650, 65295, 40355, 24940, 15415, 30830, 19055, 38110, 76220, 152440, 304880, 188425, 376850, 232905, 465810, 931620, 1863240, 1151545, 711695, 1423390, 879705, 1759410, 1087375, 672035, 415340, 256695, 158645, 317290, 196095, 121195, 242390, 149805, 92585, 57220, 35365, 70730, 43715, 87430, 54035, 33395, 20640, 12755, 25510, 15765, 9745, 19490, 12045, 7445, 4600, 9200, 5685, 3515, 7030, 4345, 2685, 1660, 1025, 635, 1270, 785, 485, 300, 185, 115, 70, …

It is interesting that the Lucas representation of 70 = 47+18+4+1, which is similar to that of 23 = 18+4+1.

[Allan Wechsler]

Geoffrey,

These examples are indeed quite eye-popping.

I take it that nothing comparably interesting happens when you use the standard Zeckendorf expansion?

-- Allan

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[Geoffrey Caveney]

Allan, I'm glad that you also find these long hailstone sequence cycles interesting.

In fact, myself and Haoxuan (Jason) Dong are also investigating the analogous function with the standard Fibonacci-Zeckendorf representation. There are quite interesting phenomena there as well, but nothing as immediately stunning as these long cycles of 23 and 70 in the Lucas-Collatz function. It appears by contrast that in the Fibonacci-Zeckendorf variant, the orbits of all integers reach the simple cycle 1, 2, 1, .... Of course we do not have a proof of this conjecture.

Geoffrey

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[Allan Wechsler]

One thing that would really help to think about this is a procedure for doubling a number in Zeckendorf notation -- I suspect that the corresponding procedure for doubling a number in the corresponding Lucas notation would be identical except for some startup transients.

The obvious procedure is to decode the Zeckendorf representation, double the number in a more amenable representation, and then re-encode it. But I think that we might be able to improve this to a state machine, where the machine receives a Zeckendorf representation one bit at a time starting from the least significant end, and outputs results one bit at a time (delayed by three or four bits). My initial fiddling suggests this could be done with on the order of half a dozen states. If that's correct, then probably a "Lucas representation" doubler could be achieved just by picking a different start state (possibly a new, bespoke state, but I suspect at most one).

-- Allan

If nobody supplies the correct procedure, I might try to work it out this evening.

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[Geoffrey Caveney]

Allan,

It appears that the paper "Complexity of Linear Subsequences of Fibonacci-Automatic Sequences", published on arXiv by Moradi, Rampersad, and Shallit in March of this year, addresses this question:

https://arxiv.org/abs/2603.21645

The abstract states:"We construct automata with input(s) in Fibonacci representation (also known as Zeckendorf representation) recognizing some basic arithmetic relations and study their number of states. We also consider some basic operations on Fibonacci-automatic sequences and discuss their state complexity. Furthermore, as a consequence of our results, we improve a bound in a recent paper of Bosma and Don. We also discuss the state complexity and runtime complexity of using a reasonable interpretation of Büchi arithmetic to actually construct some of the studied automata recognizing relations."

Section 3 "Recognizing Relations" and 3.2 "Multiplication" on pp. 8-12 address your question. The authors even draw a nice diagram on p. 9 with the caption "Figure 2: Multiplication by 2 in Fibonacci representation."

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[Jeffrey Shallit]

You can very easily get the automata for arithmetic on Lucas representations with Walnut.  The idea is to have a Zeckendorf-to-Lucas automaton A, so that one can start with a Lucas representation, do the conversion backwards with A, do the arithmetic in Zeckendorf (built in to Walnut), then convert back with A.  This gives the minimal automaton.  

On Tue, Aug 25, 2026 at 1:27 PM Allan Wechsler <acw...@gmail.com> wrote:

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[Geoffrey Caveney]

I observe that the numbers I define as "Lucas-odd" are almost an exact match with the sequence A094350, which is based on sequence A094291. (A094350 is defined as "Numbers n such that A094291(n) is not a square.") However, there is one term of A094350 which is not Lucas-odd by my definition: 41. It is not easy to double-check this value, because A094291 only lists values up to a(30). Could someone who is familiar with the subject matter of these sequences double-check the accuracy of the term 41 in A094350?

Thanks,

Geoffrey

[Hugo Pfoertner]

The first terms of https://oeis.org/A094350, together with the terms of https://oeis.org/A094291 are:

[2, 2], [5, 18], [6, 40], [9, 525], [13, 17640], [16, 261360], [17, 637065], [20, 10020010], [23, 155739584], [24, 393853824], [27, 6347376360], [31, 261312846300], [34, 4273916775040], [35, 10851741811625], [38, 181162302621300], [41, 2992431188360250], [42, 7709033255332500], [45, 129416878280289600], ...

so 41 is confirmed as a term in A094350.

[Geoffrey Caveney]

Thank you Hugo!

Geoffrey 

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[Geoffrey Caveney]

Many further thanks to Hugo for computing the terms of A094350 up to 158. The only two terms in A094350 that are not "Lucas-odd" numbers are 41 and 117, which is the Lucas number 76 + 41. Something interesting is going on here, but I don't understand what it is. The minimal Lucas representation of 41 is 10100010. Lucas-odd numbers have representations ending in 1 or in 1010.

Geoffrey

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[Hugo Pfoertner]

I'm currently calculating a b-file for A094291. It will have approximately 2000 terms before the term length limit of 1000 digits is exceeded. This will also allow for more terms of A094350. It will take a little while, though.

[Geoffrey Caveney]

Thanks to Hugo's extremely useful calculation of the first 2400 terms of A094291, I am able to make the following empirical observations about the apparent relationship between the terms of A094350, the numbers n such that A094291(n) is not a square, and the minimal Lucas representation of n:

In addition to containing the numbers with Lucas representations ending in 1 or in 1010, A094350 also includes the numbers with Lucas representations ending in 10100010, such as 41, 117, 164, .... Generalizing this pattern, we may also consider those numbers with Lucas representations ending in 101000100010, 1010001000100010, etc., i.e., 1010 followed by repeated blocks [0010]. Indeed the following calculations support this conjecture:

101000100010 = 287, and A094291(287) is not a square: its final decimal digits 85 show this immediately.

1010001000100010 = 1973, and A094291(1973) is not a square. Even though this 822-digit number is too large for WolframAlpha to process, its final decimal digits are 20000. So A094291(1973) / 10,000 cannot be a square, since no square number ends in the decimal digit 2. But 10,000 = 100^2. So A094291(1973) = 100^2 * a non-square, and thus it cannot be a square.

Geoffrey

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[h...@crypt.org]

I have limited time (and skill) to look into this deeply right now, but I
note that the combinatorial source of A094291 is fairly straightforward:

a(n) = maximal value of C(i, j) * C(n-j, n-i) for 0 <= j <= i <= n.

Is someone able to derive more about what it means for A094291(n) to be
square?

I can imagine, for example, that the matches with the minimal Lucas
representations may occur for some predictable cases where
C(i, j) = C(n-j, n-i), while the exceptions occur when the squareness
is in some sense more accidental.

Hugo ("the other Hugo") van der Sanden

Geoffrey Caveney <geoffre...@gmail.com> wrote:
:Thanks to Hugo's extremely useful calculation of the first 2400 terms of

:A094291, I am able to make the following empirical observations about the
:apparent relationship between the terms of A094350, the numbers n such
:that A094291(n)
:is not a square, and the minimal Lucas representation of n:
:
:In addition to containing the numbers with Lucas representations ending in
:1 or in 1010, A094350 also includes the numbers with Lucas representations
:ending in 10100010, such as 41, 117, 164, .... Generalizing this pattern,
:we may also consider those numbers with Lucas representations ending in
:101000100010, 1010001000100010, etc., i.e., 1010 followed by repeated
:blocks [0010]. Indeed the following calculations support this conjecture:
:

:101000100010 =3D 287, and A094291(287) is not a square: its final decimal

:digits 85 show this immediately.
:

:1010001000100010 =3D 1973, and A094291(1973) is not a square. Even though

:this 822-digit number is too large for WolframAlpha to process, its final
:decimal digits are 20000. So A094291(1973) / 10,000 cannot be a square,

:since no square number ends in the decimal digit 2. But 10,000 =3D
:100^2. So A094291(1973)
:=3D 100^2 * a non-square, and thus it cannot be a square.
:
:Geoffrey
:
:
:On Wed, Aug 26, 2026 at 12:55=E2=80=AFPM Hugo Pfoertner <yae...@gmail.com>=
: wrote:
:
:> I'm currently calculating a b-file for A094291. It will have approximatel=
:y
:> 2000 terms before the term length limit of 1000 digits is exceeded. This

:> will also allow for more terms of A094350. It will take a little while,
:> though.
:>

:> On Wednesday, August 26, 2026 at 6:00:17=E2=80=AFPM UTC+2 geoffre...@gmai=
:l.com

[Geoffrey Caveney]

Hugo V.,

As you point out, A094291(n) is predictably a square when i + j = n because one is multiplying the binomial coefficient C(i, j) by itself. There are also a few cases where i + j < n (typically, i + j = n - 1), but the product of the two distinct binomial coefficients C(i, j) * C(n-j, n-i) = A094291(n) still results in a square. I observe yet another pattern in the minimal Lucas representations of these values of n, related to but distinct from the other patterns noted previously:

n=1, i=0, j=0, 1 = 10_L

n=12, i=8, j=3, 12 = 11 + 1 = 100010_L

n=88, i=63, j=24, 88 = 76 + 11 + 1 = 1000100010_L

I also expect that n=609 belongs in this set, since the very large number A094291(609) ends in the decimal digits 0625, which is consistent with a value of a square.

n=609, i=440, j=168, 609 = 521 + 76 + 11 + 1 = 10001000100010_L

Geoffrey

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[Geoffrey Caveney]

The numbers that I describe as "Lucas-even" are now published as sequence A399305.

The numbers that I describe as "Lucas-odd" are now published as sequence A399306.

The long periodic cycle of 23 containing 132 integers in the "Lucas-Collatz" function that I describe is now published as sequence A399307. It includes a link to a b-file listing all 132 integers in the periodic cycle of 23.

I also intend to submit a similar entry for the long periodic cycle of 70 containing 122 integers, which is allocated to be sequence A399308.

Thanks to all the OEIS editors for their assistance in making the definitions and descriptions of these sequences precise.

Geoffrey

On Mon, Aug 24, 2026 at 7:35 PM Geoffrey Caveney <geoffre...@gmail.com> wrote:

[Allan Wechsler]

Geoffrey,

Shouldn't we add the "underlying" sequence, the function that is being iterated in A399307? That is, A(n) = Lucas-deflation of n if n is Lucas-even; 2n if n is Lucas-odd? If my hurried hand-calculation is correct, this begins 2,4,1,3,10,12,4,6,18,5,...

I also feel like the corresponding Zeckendorf version should be included, A(n) = Zeckendorf-deflation of n if n is Zeckendorf-even, 2n if n is Zeckendorf-odd. Again, hurried hand-calculation gives 2,1,2,8,3,12,4,5,18,6,...

If I haven't made stupid arithmetic errors, neither of these sequences are currently described in OEIS. In general, I would think that sequences related to Collatz-like processes should at minimum include the defining function itself, serving the same purpose as oeis.org/A006370 does for the OG Collatz process.

I also point out that both the Fibonacci and Lucas versions of the processes you suggest have "duals". For the Fibonacci version, it would be: n/2 if n is even; Fibonacci-inflation if n is odd. This would start 2,1,5,2,8,3,11,4,15,5,... which looks alarmingly like the "telescoped" version of the Collatz process because of the coincidence that the golden ratio is close to 3/2.

-- Allan

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[Geoffrey Caveney]

Allan, thank you very much for the feedback. That's a very good idea and I agree completely that the sequence of underlying function values should be added, for both the Lucas and Zeckendorf variants.

Geoffrey

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[Geoffrey Caveney]

Following Allan's recommendation, I have submitted the underlying sequence for the Lucas-Collatz function as draft sequence A399721.

Regarding the "dual" functions that Allan proposes -- n/2 if n is even; Zeckendorf or Lucas inflation if n is odd -- I observe that they appear to be more "well-behaved" than related sequences. For example, in the dual of the Lucas-Collatz function, 23 and 70 do not have their own periodic cycles, but reduce quickly to the basic small cycles:

23, 39, 62, 31, 48, 24, 12, 6, 3, 4, 2, 1, 3, 4, 2, 1, ...

70, 35, 55, 90, 45, 75, 120, 60, 30, 15, 25, 40, 20, 10, 5, 10, 5, ...

Likewise, despite the similar appearance of the Zeckendorf dual and the telescoped original Collatz function that Allan notes, in the dual 27 does not have a long orbit but reduces quickly to 1:

27, 44, 22, 11, 18, 9, 15, 24, 12, 6, 3, 5, 8, 4, 2, 1, 2, 1, ...

Geoffrey

On Sat, Sep 5, 2026 at 6:17 PM Allan Wechsler <acw...@gmail.com> wrote:

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