cp's OEIS Frontend

A205842 Numbers k for which 3 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 A205842 #5 Mar 30 2012 18:58:12
%S A205842 4,5,5,6,7,8,8,9,9,9,10,10,10,11,11,12,12,12,12,13,13,13,13,13,14,14,
%T A205842 14,14,15,15,15,16,16,16,16,16,17,17,17,17,17,17,18,18,18,18,18,18,19,
%U A205842 19,19,19,20,20,20,20,20,20,20,21
%N A205842 Numbers k for which 3 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 A205842 For a guide to related sequences, see A205840.
%e A205842 The first six terms match these differences:
%e A205842 s(4)-s(2) = 5-2 = 3
%e A205842 s(5)-s(2) = 8-2 = 6
%e A205842 s(5)-s(4) = 8-5 = 3
%e A205842 s(6)-s(1) = 13-1 = 12
%e A205842 s(7)-s(3) = 21-3 = 18
%e A205842 s(8)-s(1) = 34-1 = 33
%t A205842 s[n_] := s[n] = Fibonacci[n + 1]; z1 = 400; z2 = 60;
%t A205842 f[n_] := f[n] = Floor[(-1 + Sqrt[8 n - 7])/2];
%t A205842 Table[s[n], {n, 1, 30}]
%t A205842 u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
%t A205842 Table[u[m], {m, 1, z1}]   (* A204922 *)
%t A205842 v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
%t A205842 w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
%t A205842 d[n_] := d[n] = Delete[w[n], Position[w[n], 0]]
%t A205842 c = 3; t = d[c]       (* A205841 *)
%t A205842 k[n_] := k[n] = Floor[(3 + Sqrt[8 t[[n]] - 1])/2]
%t A205842 j[n_] := j[n] = t[[n]] - f[t][[n]] (f[t[[n]]] + 1)/2
%t A205842 Table[k[n], {n, 1, z2}]      (* A205842 *)
%t A205842 Table[j[n], {n, 1, z2}]      (* A205843 *)
%t A205842 Table[s[k[n]] - s[j[n]], {n, 1, z2}] (* A205844 *)
%t A205842 Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}] (* A205845 *)
%Y A205842 Cf. A204892, A205845.
%K A205842 nonn
%O A205842 1,1
%A A205842 _Clark Kimberling_, Feb 01 2012