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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.
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%I #6 Mar 30 2012 18:58:11

%S 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 12,12,12,12,12,13,13,13,13,13,13,13,13,14,14,14,14,15,15,15,15,15,15,

%U 15,15,15,16,16,16,16,16

%N 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 For a guide to related sequences, see A205840.

%e The first six terms match these differences:

%e s(3)-s(1) = 3-1 = 2

%e s(4)-s(1) = 5-1 = 4

%e s(4)-s(3) = 5-3 = 2

%e s(5)-s(2) = 8-2 = 6

%e s(6)-s(1) = 13-1 = 12

%e s(6)-s(3) = 13-3 = 10

%t s[n_] := s[n] = Fibonacci[n + 1]; z1 = 400; z2 = 60;

%t f[n_] := f[n] = Floor[(-1 + Sqrt[8 n - 7])/2];

%t Table[s[n], {n, 1, 30}]

%t u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]

%t Table[u[m], {m, 1, z1}] (* A204922 *)

%t v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]

%t w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]

%t d[n_] := d[n] = Delete[w[n], Position[w[n], 0]]

%t c = 2; t = d[c] (* A205556 *)

%t k[n_] := k[n] = Floor[(3 + Sqrt[8 t[[n]] - 1])/2]

%t j[n_] := j[n] = t[[n]] - f[t][[n]] (f[t[[n]]] + 1)/2

%t Table[k[n], {n, 1, z2}] (* A205837 *)

%t Table[j[n], {n, 1, z2}] (* A205838 *)

%t Table[s[k[n]] - s[j[n]], {n, 1, z2}](* A205839 *)

%t Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}](* A205840 *)

%Y Cf. A204892, A205840, A205558.

%K nonn

%O 1,1

%A _Clark Kimberling_, Feb 01 2012