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A205837 Numbers k for which 2 divides s(k)-s(j) for some j each k occurs once for each such j; s(k) denotes the (k+1)-st Fibonacci number.

%I A205837 #6 Mar 30 2012 18:58:11
%S A205837 3,4,4,5,6,6,6,7,7,7,7,8,8,9,9,9,9,9,10,10,10,10,10,10,11,11,11,12,12,
%T A205837 12,12,12,12,12,13,13,13,13,13,13,13,13,14,14,14,14,15,15,15,15,15,15,
%U A205837 15,15,15,16,16,16,16,16
%N A205837 Numbers k for which 2 divides s(k)-s(j) for some j<k; each k occurs once for each such j; s(k) denotes the (k+1)-st Fibonacci number.
%C A205837 For a guide to related sequences, see A205840.
%e A205837 The first six terms match these differences:
%e A205837 s(3)-s(1) = 3-1 = 2
%e A205837 s(4)-s(1) = 5-1 = 4
%e A205837 s(4)-s(3) = 5-3 = 2
%e A205837 s(5)-s(2) = 8-2 = 6
%e A205837 s(6)-s(1) = 13-1 = 12
%e A205837 s(6)-s(3) = 13-3 = 10
%t A205837 s[n_] := s[n] = Fibonacci[n + 1]; z1 = 400; z2 = 60;
%t A205837 f[n_] := f[n] = Floor[(-1 + Sqrt[8 n - 7])/2];
%t A205837 Table[s[n], {n, 1, 30}]
%t A205837 u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
%t A205837 Table[u[m], {m, 1, z1}]   (* A204922 *)
%t A205837 v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
%t A205837 w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
%t A205837 d[n_] := d[n] = Delete[w[n], Position[w[n], 0]]
%t A205837 c = 2; t = d[c]           (* A205556 *)
%t A205837 k[n_] := k[n] = Floor[(3 + Sqrt[8 t[[n]] - 1])/2]
%t A205837 j[n_] := j[n] = t[[n]] - f[t][[n]] (f[t[[n]]] + 1)/2
%t A205837 Table[k[n], {n, 1, z2}]     (* A205837 *)
%t A205837 Table[j[n], {n, 1, z2}]     (* A205838 *)
%t A205837 Table[s[k[n]] - s[j[n]], {n, 1, z2}](* A205839 *)
%t A205837 Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}](* A205840 *)
%Y A205837 Cf. A204892, A205840, A205558.
%K A205837 nonn
%O A205837 1,1
%A A205837 _Clark Kimberling_, Feb 01 2012