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A205865 [s(k)-s(j)]/7, where the pairs (k,j) are given by A205862 and A205863, and s(k) denotes the (k+1)-st Fibonacci number.

%I A205865 #5 Mar 30 2012 18:58:12
%S A205865 1,3,6,3,12,33,52,49,46,87,86,138,228,227,141,369,368,282,141,597,564,
%T A205865 966,1563,1551,2530,2529,2443,2302,2161,4092,4089,4086,4040,6621,6483,
%U A205865 10716,10713,10710,10664,6624,17340,17337,17334,17288,13248
%N A205865 [s(k)-s(j)]/7, where the pairs (k,j) are given by A205862 and A205863, and s(k) denotes the (k+1)-st Fibonacci number.
%C A205865 For a guide to related sequences, see A205840.
%e A205865 The first six terms match these differences:
%e A205865 s(5)-s(1) = 8-1 = 7 = 7*1
%e A205865 s(8)-s(6) = 34-13 = 21 = 7*3
%e A205865 s(9)-s(6) = 55-13 = 42 = 7*6
%e A205865 s(9)-s(8) = 55-34 = 21 = 7*3
%e A205865 s(10)-s(4) = 89-5 = 84 = 7*12
%e A205865 s(13)-s(6) = 377-13 = 364 =7*52
%t A205865 s[n_] := s[n] = Fibonacci[n + 1]; z1 = 500; z2 = 60;
%t A205865 f[n_] := f[n] = Floor[(-1 + Sqrt[8 n - 7])/2];
%t A205865 Table[s[n], {n, 1, 30}]
%t A205865 u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
%t A205865 Table[u[m], {m, 1, z1}]  (* A204922 *)
%t A205865 v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
%t A205865 w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
%t A205865 d[n_] := d[n] = Delete[w[n], Position[w[n], 0]]
%t A205865 c = 7; t = d[c]   (* A205861 *)
%t A205865 k[n_] := k[n] = Floor[(3 + Sqrt[8 t[[n]] - 1])/2]
%t A205865 j[n_] := j[n] = t[[n]] - f[t][[n]] (f[t[[n]]] + 1)/2
%t A205865 Table[k[n], {n, 1, z2}]    (* A205862 *)
%t A205865 Table[j[n], {n, 1, z2}]    (* A205863 *)
%t A205865 Table[s[k[n]] - s[j[n]], {n, 1, z2}]  (* A205864 *)
%t A205865 Table[(s[k[n]]-s[j[n]])/c, {n,1,z2}]  (* A205865 *)
%Y A205865 Cf. A204892, A205862, A205864.
%K A205865 nonn
%O A205865 1,2
%A A205865 _Clark Kimberling_, Feb 02 2012