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A205860 [s(k)-s(j)]/6, where the pairs (k,j) are given by A205857 and A205858, and s(k) denotes the (k+1)-st Fibonacci number.

%I A205860 #5 Mar 30 2012 18:58:12
%S A205860 1,2,3,9,7,14,38,24,62,48,24,96,164,161,266,264,257,425,329,696,682,
%T A205860 658,634,1127,1124,963,1824,1823,2951,2937,2913,2889,2255,4776,4774,
%U A205860 4767,4510,7704,12504,12502,12495,12238,7728,20232,20230,20223
%N A205860 [s(k)-s(j)]/6, where the pairs (k,j) are given by A205857 and A205858, and s(k) denotes the (k+1)-st Fibonacci number.
%C A205860 For a guide to related sequences, see A205840.
%e A205860 The first six terms match these differences:
%e A205860 s(5)-s(2) = 8-2 = 6 = 6*1
%e A205860 s(6)-s(1) = 13-1 = 12 = 6*2
%e A205860 s(7)-s(3) = 21-3 = 18 = 6*3
%e A205860 s(9)-s(1) = 55-1 = 54 = 6*9
%e A205860 s(9)-s(6) = 55-13 = 42 = 6*7
%e A205860 s(10)-s(4) = 89-5 = 84 =6*14
%t A205860 s[n_] := s[n] = Fibonacci[n + 1]; z1 = 500; z2 = 60;
%t A205860 f[n_] := f[n] = Floor[(-1 + Sqrt[8 n - 7])/2];
%t A205860 Table[s[n], {n, 1, 30}]
%t A205860 u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
%t A205860 Table[u[m], {m, 1, z1}]   (* A204922 *)
%t A205860 v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
%t A205860 w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
%t A205860 d[n_] := d[n] = Delete[w[n], Position[w[n], 0]]
%t A205860 c = 6; t = d[c]    (* A205856 *)
%t A205860 k[n_] := k[n] = Floor[(3 + Sqrt[8 t[[n]] - 1])/2]
%t A205860 j[n_] := j[n] = t[[n]] - f[t][[n]] (f[t[[n]]] + 1)/2
%t A205860 Table[k[n], {n, 1, z2}]     (* A205857 *)
%t A205860 Table[j[n], {n, 1, z2}]     (* A205858 *)
%t A205860 Table[s[k[n]]-s[j[n]], {n, 1, z2}]    (* A205859 *)
%t A205860 Table[(s[k[n]]-s[j[n]])/c, {n,1,z2}]  (* A205860 *)
%Y A205860 Cf. A204892, A205857, A205859.
%K A205860 nonn
%O A205860 1,2
%A A205860 _Clark Kimberling_, Feb 02 2012