RFE Mar 2026: Soddy's bowl of integers

[Sean A. Irvine]

Hi,

This month I have a request that might offer an opportunity to do some
actual math:

The "nice" pair of sequences A046159 and A046160 correspond to placing
progressively smaller spheres in a particular arrangement. The
sequences are based on a short 1937 paper (linked in the sequences)
from Frederick Soddy. As usual, I would like to see more terms,
formulas, explanations, and/or programs in the OEIS sequences. This
problem is potentially interesting enough to yield some new sequences
as well.

Thanks to Thomas Scheurle and Natalia L. Skirrow who helped resolve
last month's request.

I track these requests for enhancement here:

https://oeis.org/wiki/User:Sean_A._Irvine/Requests_for_Enhancements#Requests_for_Enhancements

Sean.

[Martin Fuller]

I am fairly sure that A046159 can be generated with the following PARI program:

bend(K)=my(s=vecsum(K),ss=vecsum(apply(k->k^2,K)),n=#K-1);(s+sqrtint(n*s^2-n*(n-1)*ss))/(n-1);
a(limit)=
{
  my(s=Set, v=[3]);
  while(#s < limit || s[limit] > vecmin(v),
    s = setunion(s, Set(v));
    my(w=vector(#v, i, bend([2, 2, v[i], v[(i%#v)+1]])));
    v = concat(bend([2, 2, v[1], w[1]]), w);
  );
  vecextract(s, [1..limit])
};

It is based on an observation about the second set of diagrams at the Mathworld link which together show the spheres in the plane. The spheres are arranged in concentric circles with sixfold symmetry, first 6*[3] on the outside then 6*[15,11], then 6*[35,27,27], 6*[63,51,47,51] etc. In each case the repeating pattern has one sphere that touches just one from the previous ring, and the remainder touch two each. The single-touch spheres form 6 lines through the origin. The program uses this arrangement and Soddy's formula to calculate bends in each ring, until the values are too big and it can stop.  For the formula see https://mathworld.wolfram.com/TangentSpheres.html equation (6) or (13).

It appears (conjecture) that the single-touch spheres match https://oeis.org/A000466 including -1 for the outermost sphere.

Proof that there is one single-touch sphere per repeating pattern, and they are connected:

This is an application of the Hexlet theorem or Kollros' Theorem (see Mathworld). Consider any sphere in the plane and the two "2" spheres: these form a ring of 3 spheres. Therefore there is another ring of 6 spheres which touch the original 3, and they are all in the plane. Two of these six must be from the same concentric circle as the original. If there is a single-touch sphere from the outer circle then there must be three in the inner circle, one of which is single-touch. If there are two touching spheres from the outer circle then there must be two from the inner circle, and none of these are single-touch.

I hope that makes sense.

Martin Fuller

[Thomas Scheuerle]

It appears that it shares many values with A270248 - 1. Purely a coincidence ?

[Martin Fuller]

On Thursday, 5 March 2026 at 12:45:09 UTC Thomas Scheuerle wrote:

It appears that it shares many values with A270248 - 1. Purely a coincidence ?

No!

Claim: the bends in concentric circle n=1,2... are the values 4(x^2+xy+y^2)-1 for x+y=n with x=0..n-1 repeated six times.
The spheres with x=0 have one outer neighbour and 3 inner neighbours. Other spheres have two of each.

Proof by induction on n:
The bend for a sphere touching 2,2,K1,K2 is a specialization of Soddy's formula:
K(K1,K2) = (4+K1+K2 + sqrt(3*(4+K1+K2)^2 - 6*(8+K1^2+K2^2))) / 2.

Define (x,y) = 4(x^2+xy+y^2)-1.

Take neighbouring spheres in circle n. By induction hypothesis K1 = (x,y), K2 = (x+1,y-1).
It is safe to use x=n-1 because K2=(n,0)=(0,n) is the first sphere of the next repetition.
The bend of the inner sphere touching 2,2,K1,K2 works out to be K(K1,K2) = (x+1,y).

There is one extra sphere in ring n+1 which touches (0,n) and (1,n), as described in my previous post.
The bend of this sphere works out to be K((0,n), (1,n)) = (0,n+1).

Corollary: Spheres on the six radial lines have bends given by (0,n) = 4n^2-1 = A000466(n).

A270248 starts with a(1)=0, which corresponds to the surrounding unit sphere with bend -1.

Martin

PS Does A046160 list spheres that are not in the plane?

[Martin Fuller]

Does the bowl contain spheres that are not in contact with the original pair of 2-spheres?

Extract from Soddy's paper:

"So we may imagine similar rings of spheres above and below the plane of the paper,

each in contact with the bowl and with one of the two 2-spheros, and then begin all over again

to fill up the remaining spaces and so on ad infinitum, every sphere added increasing the

number that have to be added to fill it up !

These further spheres being spaced in three dimensions cannot be accurately represented by a drawing, ..."

The analogous problem with circles, from the end of the Mathworld link, does fill all the gaps.

Does this continue for spheres? If yes, are any of these extra spheres in the plane?

Should A046159 and A046160 cover all spheres or only those that contact the original 2-spheres?

On Monday, 2 March 2026 at 20:32:51 UTC sai...@gmail.com wrote:

[Martin Fuller]

Suggestion: A046160 should be "Bends of spheres in Soddy's bowl of integers which touch the unit sphere and a 2-sphere and are not in the plane."

This corresponds roughly to Soddy's Fig 2, and to this text opposite the figure:

  "successive rings I, 3 ;  II, 6, 5 ;  III, 11, 9, 9 ;  IV, 18, 15, 14, 15 ;  V, 27, 23, 21, 21, 23 ;  VI, 38, 33, 30, 29, 30, 33 and so on."

The description tries to justify why the sequence starts with 2 instead of 3. I don't know why the existing sequence skips 29.

This sequence continues: 2, 5, 6, 9, 11, 14, 15, 18, 21, 23, 27, 29, 30, 33, 38, 39, 41, 45, 50, 51, 54, 59, 63, 65, 66, 69, 75, 77, 78, 81, 83, 86, 93, 95, 99, 102, 105, 110, 111, 113, 114, 119, 123, 126, 129, 131, 135, 141, 146, 149, 150, 153, 158, 159, 165, 171, 173, 174, 177, 183.

Conjecture: The bowl has 60-degree lines of longitude which are tangent to spheres with bends given by A059100(n) = n^2 + 2.

Crossrefs:

The list of all bends in the bowl is given by A045506.

* The smallest number in A045506 not in A046159 or A046160 is 8 which is tangent to (5,3,3,-1).

* This is not shown in Fig. 2 or Mathworld first image. It would be added on the outside near the equator.

Conjecture: the list of all bends not in the plane is A049636 "Congruent to 0 or 2 mod 3, but not equal to 0 or 3."

On Monday, 2 March 2026 at 20:32:51 UTC sai...@gmail.com wrote: