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A205872 Numbers k for which 9 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 A205872 #6 Mar 30 2012 18:58:12
%S A205872 7,9,10,12,12,13,13,13,14,16,17,17,18,19,20,21,21,21,21,22,22,22,23,
%T A205872 24,24,24,24,25,25,25,25,25,26,26,27,27,28,28,29,29,29,29,29,30,30,31,
%U A205872 31,31,32,32
%N A205872 Numbers k for which 9 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 A205872 For a guide to related sequences, see A205840.
%e A205872 The first six terms match these differences:
%e A205872 s(7)-s(3) = 21-3 = 18 = 9*2
%e A205872 s(9)-s(1) = 55-1 = 54 = 9*6
%e A205872 s(10)-s(5) = 89-8 = 81 = 9*9
%e A205872 s(12)-s(5) = 233-8 = 225 = 9*25
%e A205872 s(12)-s(10) = 233-89 = 144 = 9*16
%e A205872 s(13)-s(5) = 377-8 = 369 =9*41
%t A205872 s[n_] := s[n] = Fibonacci[n + 1]; z1 = 600; z2 = 50;
%t A205872 f[n_] := f[n] = Floor[(-1 + Sqrt[8 n - 7])/2];
%t A205872 Table[s[n], {n, 1, 30}]
%t A205872 u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
%t A205872 Table[u[m], {m, 1, z1}]   (* A204922 *)
%t A205872 v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
%t A205872 w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
%t A205872 d[n_] := d[n] = Delete[w[n], Position[w[n], 0]]
%t A205872 c = 9; t = d[c]     (* A205871 *)
%t A205872 k[n_] := k[n] = Floor[(3 + Sqrt[8 t[[n]] - 1])/2]
%t A205872 j[n_] := j[n] = t[[n]] - f[t][[n]] (f[t[[n]]] + 1)/2
%t A205872 Table[k[n], {n, 1, z2}]       (* A205872 *)
%t A205872 Table[j[n], {n, 1, z2}]         (* A205873 *)
%t A205872 Table[s[k[n]] - s[j[n]], {n, 1, z2}]   (* A205874 *)
%t A205872 Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}]  (* A205875 *)
%Y A205872 Cf. A204892, A205873, A205875.
%K A205872 nonn
%O A205872 1,1
%A A205872 _Clark Kimberling_, Feb 02 2012