A205870 - OEIS

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

%S 1,2,1,4,11,17,29,18,47,36,18,76,72,123,199,198,197,322,305,522,521,

%T 520,323,845,844,843,646,323,1368,1364,1292,2207,3582,3571,3553,3535,

%U 5795,5778,5473,9378,9367,9349,9331,5796,15174,15163,15145,15127

%N [s(k)-s(j)]/8, where the pairs (k,j) are given by A205867 and A205868, 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(6)-s(4) = 13-5 = 8 = 8*1

%e s(7)-s(4) = 21-5 = 16 = 8*2

%e s(7)-s(6) = 21-13 = 8 = 8*1

%e s(8)-s(2) = 34-2 = 32 = 8*4

%e s(10)-s(1) = 89-1 = 88 = 8*11

%e s(11)-s(5) = 144-8 = 136 =8*17

%t s[n_] := s[n] = Fibonacci[n + 1]; z1 = 600; z2 = 50;

%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 = 8; t = d[c]    (* A205866 *)

%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}]   (* A205867 *)

%t Table[j[n], {n, 1, z2}]     (* A205868 *)

%t Table[s[k[n]] - s[j[n]], {n, 1, z2}] (* A205869 *)

%t Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}]  (* A205870 *)

%Y Cf. A204892, A205867, A205869.

%K nonn

%O 1,2

%A _Clark Kimberling_, Feb 02 2012