A395251 a(n) gives positions k >= 1 where the equation b(b(k)+b(k-1)) = k fails, for b(k) = floor(k/sqrt(2) + 2*(sqrt(2)-1)).

6, 9, 13, 16, 23, 30, 33, 40, 47, 50, 54, 57, 64, 67, 71, 74, 81, 88, 91, 95, 98, 105, 108, 112, 115, 122, 129, 132, 139, 146, 149, 153, 156, 163, 170, 173, 180, 187, 190, 194, 197, 204, 207, 211, 214, 221, 228, 231, 238, 245, 248, 252, 255, 262, 269, 272, 279, 286, 289, 293, 296

Offset

1,1

Comments

The Beatty sequence b(k) = floor(k/sqrt(2) + 2*(sqrt(2)-1)) satisfies the triple-nested equation b(b(b(k)+b(k-1))) = b(k) for all k >= 1 (see reference).

Equivalent characterization: k is in this sequence iff frac((k+3)/sqrt(2)) lies in [1 - 1/sqrt(2), 1/2].

The sequence of differences a(n+1) - a(n) takes values in {3, 4, 7} and belongs to the minimal subshift generated by the primitive substitution 3 -> 3,4; 4 -> 3,7; 7 -> 3,7,7 (see A395253), with Perron eigenvalue 1 + sqrt(2).

Links

Table of n, a(n) for n=1..61.

Benoit Cloitre, Beatty solutions of almost Golomb equations, arXiv:2604.10822 [math.NT], 2026.

Formula

a(n) ~ 2*(sqrt(2)+1)*n as n --> infinity.

Mathematica

b[k_] := Floor[k/Sqrt[2] + 2*(Sqrt[2] - 1)]; q[k_] := b[b[k] + b[k - 1]] != k; Select[Range[300], q] (* Amiram Eldar, Apr 17 2026 *)

Crossrefs

Cf. A001951, A001952, A001333, A000129, A394217, A395252, A395253.

Sequence in context: A284800 A242295 A064799 * A236763 A030499 A181428

Adjacent sequences: A395247 A395248 A395250 * A395252 A395253 A395254

Keyword

nonn

Author

Benoit Cloitre, Apr 17 2026

Status

approved