A205870 [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.

1, 2, 1, 4, 11, 17, 29, 18, 47, 36, 18, 76, 72, 123, 199, 198, 197, 322, 305, 522, 521, 520, 323, 845, 844, 843, 646, 323, 1368, 1364, 1292, 2207, 3582, 3571, 3553, 3535, 5795, 5778, 5473, 9378, 9367, 9349, 9331, 5796, 15174, 15163, 15145, 15127

Offset: 1

Author

Clark Kimberling, Feb 02 2012

Keywords

• nonn

Comments

For a guide to related sequences, see A205840.

Examples

			The first six terms match these differences:
s(6)-s(4) = 13-5 = 8 = 8*1
s(7)-s(4) = 21-5 = 16 = 8*2
s(7)-s(6) = 21-13 = 8 = 8*1
s(8)-s(2) = 34-2 = 32 = 8*4
s(10)-s(1) = 89-1 = 88 = 8*11
s(11)-s(5) = 144-8 = 136 =8*17
		

Crossrefs

Cf. A204892, A205867, A205869.

Programs

Mathematica

s[n_] := s[n] = Fibonacci[n + 1]; z1 = 600; z2 = 50;
f[n_] := f[n] = Floor[(-1 + Sqrt[8 n - 7])/2];
Table[s[n], {n, 1, 30}]
u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
Table[u[m], {m, 1, z1}] (* A204922 *)
v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
d[n_] := d[n] = Delete[w[n], Position[w[n], 0]]
c = 8; t = d[c]    (* A205866 *)
k[n_] := k[n] = Floor[(3 + Sqrt[8 t[[n]] - 1])/2]
j[n_] := j[n] = t[[n]] - f[t][[n]] (f[t[[n]]] + 1)/2
Table[k[n], {n, 1, z2}]   (* A205867 *)
Table[j[n], {n, 1, z2}]     (* A205868 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}] (* A205869 *)
Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}]  (* A205870 *)