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A205852 Numbers k for which 5 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 A205852 #5 Mar 30 2012 18:58:12
%S A205852 5,6,6,7,9,10,11,11,12,12,12,13,14,14,15,15,16,16,16,17,17,17,18,18,
%T A205852 19,19,19,20,20,20,21,21,21,21,22,22,22,22,23,23,23,23,24,24,24,24,25,
%U A205852 25,25,25,25,26,26,26,26,26,26,27,27,27
%N A205852 Numbers k for which 5 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 A205852 For a guide to related sequences, see A205840.
%e A205852 The first six terms match these differences:
%e A205852 s(5)-s(3) = 8-3 = 5
%e A205852 s(6)-s(3) = 13-3 = 10
%e A205852 s(6)-s(5) = 13-8 = 5
%e A205852 s(7)-s(1) = 21-1 = 20
%e A205852 s(9)-s(4) = 55-5 = 50
%e A205852 s(10)-s(8) = 89-34 = 55
%t A205852 s[n_] := s[n] = Fibonacci[n + 1]; z1 = 500; z2 = 60;
%t A205852 f[n_] := f[n] = Floor[(-1 + Sqrt[8 n - 7])/2];
%t A205852 Table[s[n], {n, 1, 30}]
%t A205852 u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
%t A205852 Table[u[m], {m, 1, z1}]    (* A204922 *)
%t A205852 v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
%t A205852 w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
%t A205852 d[n_] := d[n] = Delete[w[n], Position[w[n], 0]]
%t A205852 c = 5; t = d[c]    (* A205851 *)
%t A205852 k[n_] := k[n] = Floor[(3 + Sqrt[8 t[[n]] - 1])/2]
%t A205852 j[n_] := j[n] = t[[n]] - f[t][[n]] (f[t[[n]]] + 1)/2
%t A205852 Table[k[n], {n, 1, z2}] (* A205852 *)
%t A205852 Table[j[n], {n, 1, z2}] (* A205853 *)
%t A205852 Table[s[k[n]]-s[j[n]], {n, 1, z2}](* A205854 *)
%t A205852 Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}] (* A205855 *)
%Y A205852 Cf. A204892, A205855.
%K A205852 nonn
%O A205852 1,1
%A A205852 _Clark Kimberling_, Feb 02 2012