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A205847 Numbers k for which 4 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 A205847 #9 Oct 25 2024 04:53:14
%S A205847 4,6,6,7,7,7,8,9,10,10,10,10,11,12,12,12,12,12,13,13,13,13,13,13,14,
%T A205847 14,15,15,16,16,16,16,16,16,16,17,17,18,18,18,18,18,18,18,18,19,19,19,
%U A205847 19,19,19,19,19,19,20,20,20,21,21,21
%N A205847 Numbers k for which 4 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 A205847 For a guide to related sequences, see A205840.
%e A205847 The first six terms match these differences:
%e A205847 s(4)-s(1) = 5-1 = 4
%e A205847 s(6)-s(1) = 13-1 = 12
%e A205847 s(6)-s(4) = 13-5 = 8
%e A205847 s(7)-s(1) = 21-1 = 20
%e A205847 s(7)-s(4) = 21-5 = 16
%e A205847 s(7)-s(6) = 21-13 = 8
%t A205847 s[n_] := s[n] = Fibonacci[n + 1]; z1 = 400; z2 = 60;
%t A205847 f[n_] := f[n] = Floor[(-1 + Sqrt[8 n - 7])/2];
%t A205847 Table[s[n], {n, 1, 30}]
%t A205847 u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
%t A205847 Table[u[m], {m, 1, z1}]     (* A204922 *)
%t A205847 v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
%t A205847 w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
%t A205847 d[n_] := d[n] = Delete[w[n], Position[w[n], 0]]
%t A205847 c = 4; t = d[c]    (* A205846 *)
%t A205847 k[n_] := k[n] = Floor[(3 + Sqrt[8 t[[n]] - 1])/2]
%t A205847 j[n_] := j[n] = t[[n]] - f[t][[n]] (f[t[[n]]] + 1)/2
%t A205847 Table[k[n], {n, 1, z2}]    (* A205847 *)
%t A205847 Table[j[n], {n, 1, z2}]    (* A205848 *)
%t A205847 Table[s[k[n]] - s[j[n]], {n, 1, z2}] (* A205849 *)
%t A205847 Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}] (* A205850 *)
%Y A205847 Cf. A204892, A205850.
%K A205847 nonn
%O A205847 1,1
%A A205847 _Clark Kimberling_, Feb 02 2012