A046159 Bends of spheres in the plane of Soddy's bowl of integers.

3, 11, 15, 27, 35, 47, 51, 63, 75, 83, 99, 107, 111, 123, 143, 147, 155, 171, 191, 195, 207, 227, 243, 251, 255, 267, 291, 299, 303, 315, 323, 335, 363, 371, 387, 399, 411, 431, 435, 443, 447, 467, 483, 495, 507, 515, 531, 555, 575, 587, 591, 603, 623, 627, 651

Offset

1,1

Comments

From Martin Fuller, Mar 05 2026: (Start)

The spheres are arranged in concentric rings. Bends in ring m = 1, 2... are given by 4(x^2 + xy + y^2) - 1 with x + y = m and x = 0..m-1 repeated six times. Compare A270248. The surrounding unit sphere with bend -1 could be considered to be in ring m = 0.

There are six radial lines of touching spheres with bends given by A000466. (End)

Links

Table of n, a(n) for n=1..55.

SeqFan group, RFE Mar 2026: Soddy's bowl of integers

Frederick Soddy, The Bowl of Integers and the Hexlet, Nature 139, 77-79, 1937 [Accessible copy]

Eric Weisstein's World of Mathematics, Bowl of Integers

Formula

a(n) = A270248(n+1)-1. - Martin Fuller, Mar 05 2026

Prog

(PARI) a(limit)={
  my(s = Set);
  for(m = 1, oo,
    my(ring = Set(apply(x->my(y=m-x); 4*(x^2+x*y+y^2)-1, [0..m-1])));
    if(#s >= limit && ring[1] > s[limit], return(s[1..limit]));
    s = setunion(s, ring);
  );
}; \\ Martin Fuller, Mar 07 2026

Crossrefs

Cf. A046160, A270248, A000466.

Cf. A045506 (list of all bends: -1, 2, 3, 5, ...).

Sequence in context: A357440 A186302 A323103 * A022410 A146254 A039503

Adjacent sequences: A046156 A046157 A046158 * A046160 A046161 A046162

Keyword

nonn,nice

Author

Eric W. Weisstein

Extensions

a(30) onward from Martin Fuller, Mar 05 2026

Status

approved