A205865 - OEIS

%I #5 Mar 30 2012 18:58:12

%S 1,3,6,3,12,33,52,49,46,87,86,138,228,227,141,369,368,282,141,597,564,

%T 966,1563,1551,2530,2529,2443,2302,2161,4092,4089,4086,4040,6621,6483,

%U 10716,10713,10710,10664,6624,17340,17337,17334,17288,13248

%N [s(k)-s(j)]/7, where the pairs (k,j) are given by A205862 and A205863, and s(k) denotes the (k+1)-st Fibonacci number.

%C For a guide to related sequences, see A205840.

%e The first six terms match these differences:

%e s(5)-s(1) = 8-1 = 7 = 7*1

%e s(8)-s(6) = 34-13 = 21 = 7*3

%e s(9)-s(6) = 55-13 = 42 = 7*6

%e s(9)-s(8) = 55-34 = 21 = 7*3

%e s(10)-s(4) = 89-5 = 84 = 7*12

%e s(13)-s(6) = 377-13 = 364 =7*52

%t s[n_] := s[n] = Fibonacci[n + 1]; z1 = 500; z2 = 60;

%t f[n_] := f[n] = Floor[(-1 + Sqrt[8 n - 7])/2];

%t Table[s[n], {n, 1, 30}]

%t u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]

%t Table[u[m], {m, 1, z1}]  (* A204922 *)

%t v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]

%t w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]

%t d[n_] := d[n] = Delete[w[n], Position[w[n], 0]]

%t c = 7; t = d[c]   (* A205861 *)

%t k[n_] := k[n] = Floor[(3 + Sqrt[8 t[[n]] - 1])/2]

%t j[n_] := j[n] = t[[n]] - f[t][[n]] (f[t[[n]]] + 1)/2

%t Table[k[n], {n, 1, z2}]    (* A205862 *)

%t Table[j[n], {n, 1, z2}]    (* A205863 *)

%t Table[s[k[n]] - s[j[n]], {n, 1, z2}]  (* A205864 *)

%t Table[(s[k[n]]-s[j[n]])/c, {n,1,z2}]  (* A205865 *)

%Y Cf. A204892, A205862, A205864.

%K nonn

%O 1,2

%A _Clark Kimberling_, Feb 02 2012