A387351 Positions of -1's in A387348.

3, 8, 16, 21, 24, 29, 37, 42, 45, 50, 55, 58, 63, 71, 76, 79, 84, 92, 97, 105, 110, 113, 118, 126, 131, 134, 139, 144, 147, 152, 160, 165, 168, 173, 181, 186, 194, 199, 202, 207, 215, 220, 223, 228, 236, 241, 249, 254, 257, 262, 270, 275, 278, 283, 288, 291

Offset

1,1

Comments

This sequence together with A387349 and A387350 partition the positive integers.

Conjecture: the difference sequence, (5, 8, 5, 3, 5, 8, 5, 3, 5, 5, 3, 5, 8, 5, 3, 5, 8, 5, 8, ... ) has exactly 3 distinct terms.

These are the numbers k such that the k-th Wythoff pair does not occur as a pair of consecutive numbers in a row of the Stolarksy array; e.g., a(1) = 3 because the 3rd Wythoff pair, (4,7) is the least Wythoff pair not in a row of the Stolarsky array. - Clark Kimberling, Oct 21 2025

Numbers k such that floor(k*phi^2 - phi/2) = floor(k*phi^2). - Clark Kimberling, Jul 13 2026

Links

Table of n, a(n) for n=1..56.

Mathematica

z = 300;
A[n_, k_] := Module[{t, a, b}, t = (1 + Sqrt[5])/2;
a = Floor[n*(t + 1) + 1 + t/2]; b = Round[a*t]; ({b, a} . MatrixPower[{{1, 1}, {1, 0}}, k])[[2]]];
ts = Table[A[n, k], {n, 0, z - 1}, {k, 0, z - 1}];  (* A035506, Stolarsky array *)
W[n_, k_] := Fibonacci[k + 1] Floor[n*GoldenRatio] + (n - 1) Fibonacci[k];
tw = Table[W[n, k], {n, 1, z}, {k, 1, z}];   (* A035513, Wythoff array *)
diff = tw - ts;
u = Table[diff[[n]][[2]], {n, 1, z}]
Flatten[Position[u, 0]]   (* A387349 *)
Flatten[Position[u, 1]]   (* A387350 *)
Flatten[Position[u, -1]]  (* A387351 *)
(* Alternative: *)
r = GoldenRatio; Select[Range[1000], Floor[#*r^2] != Floor[Floor[#*r]*r + 1/2] &]
(* Clark Kimberling, Oct 21 2025 *)

Crossrefs

Cf. A387348, A387349, A387350.

Sequence in context: A368041 A030417 A275239 * A190450 A188012 A123979

Adjacent sequences: A387348 A387349 A387350 * A387352 A387353 A387354

Keyword

nonn,changed

Author

Clark Kimberling, Aug 27 2025

Status

approved