A395252 a(n) gives positions k >= 1 where the equation b(b(k)+b(k-1)+b(k-2)) = k fails, for b(k) = floor(k/sqrt(3) + (5*sqrt(3)-3)/6).

16, 23, 42, 49, 61, 68, 87, 94, 113, 120, 139, 146, 158, 165, 184, 191, 210, 217, 229, 236, 243, 255, 262, 281, 288, 307, 314, 326, 333, 340, 352, 359, 378, 385, 404, 411, 423, 430, 449, 456, 475, 482, 501, 508, 520, 527, 546, 553, 572, 579, 591, 598, 605, 617

Offset

1,1

Comments

The Beatty sequence b(k) = floor(k/sqrt(3) + (5*sqrt(3)-3)/6) satisfies the triple-nested equation b(b(b(k)+b(k-1)+b(k-2))) = b(k) for all k >= 3 (see reference).

Equivalent characterization: k is in this sequence iff frac((k-2)/sqrt(3) + (5*sqrt(3)-3)/6) lies in [0, (2-sqrt(3))/3].

a(n+1) - a(n) takes values in {7, 12, 19}.

Links

Table of n, a(n) for n=1..54.

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

Formula

a(n) ~ 3*(2+sqrt(3))*n as n --> infinity.

Mathematica

b[k_] := Floor[k/Sqrt[3] + (5*Sqrt[3] - 3)/6]; q[k_] := b[b[k] + b[k - 1] + b[k - 2]] != k; Select[Range[1000], q] (* Amiram Eldar, Apr 17 2026 *)

Crossrefs

Cf. A395251, A040001, A002531, A002530.

Sequence in context: A176657 A006616 A225929 * A179370 A166675 A114405

Adjacent sequences: A395242 A395250 A395251 * A395253 A395254 A395255

Keyword

nonn

Author

Benoit Cloitre, Apr 17 2026

Status

approved