A395251 - OEIS

%I #10 Apr 22 2026 16:47:08

%S 6,9,13,16,23,30,33,40,47,50,54,57,64,67,71,74,81,88,91,95,98,105,108,

%T 112,115,122,129,132,139,146,149,153,156,163,170,173,180,187,190,194,

%U 197,204,207,211,214,221,228,231,238,245,248,252,255,262,269,272,279,286,289,293,296

%N 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)).

%C 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).

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

%C 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).

%H Benoit Cloitre, <a href="https://arxiv.org/abs/2604.10822">Beatty solutions of almost Golomb equations</a>, arXiv:2604.10822 [math.NT], 2026.

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

%t 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 *)

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

%K nonn

%O 1,1

%A _Benoit Cloitre_, Apr 17 2026