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A387349 Positions of 0's in A387348.

%I A387349 #8 Sep 02 2025 09:24:24
%S A387349 1,4,6,7,9,11,12,14,17,19,20,22,25,27,30,32,33,35,38,40,41,43,46,48,
%T A387349 51,53,54,56,59,61,62,64,66,67,69,72,74,75,77,80,82,85,87,88,90,93,95,
%U A387349 96,98,100,101,103,106,108,109,111,114,116,117,119,121,122
%N A387349 Positions of 0's in A387348.
%C A387349 This sequence together with A387350 and A387351 partition the positive integers. Conjecture: the difference sequence (3, 2, 1, 2, 2, 1, 2, 3, 2, 1, 2, 3, 2, 3, 2, 1, ... ) has exactly 3 distinct terms.
%t A387349 z = 300;
%t A387349 A[n_, k_] := Module[{t, a, b}, t = (1 + Sqrt[5])/2;
%t A387349 a = Floor[n*(t + 1) + 1 + t/2]; b = Round[a*t]; ({b, a} . MatrixPower[{{1, 1}, {1, 0}}, k])[[2]]];
%t A387349 ts = Table[A[n, k], {n, 0, z - 1}, {k, 0, z - 1}];  (* A035506, Stolarsky array *)
%t A387349 W[n_, k_] := Fibonacci[k + 1]  Floor[n*GoldenRatio] + (n - 1)  Fibonacci[k];
%t A387349 tw = Table[W[n, k], {n, 1, z}, {k, 1, z}];   (* A035513, Wythoff array *)
%t A387349 diff = tw - ts;
%t A387349 u = Table[diff[[n]][[2]], {n, 1, z}]
%t A387349 Flatten[Position[u, 0]]   (* A387349 *)
%t A387349 Flatten[Position[u, 1]]   (* A387350 *)
%t A387349 Flatten[Position[u, -1]]  (* A387351 *)
%Y A387349 Cf. A387348, A387350, A387351.
%K A387349 nonn,new
%O A387349 1,2
%A A387349 _Clark Kimberling_, Aug 27 2025