A205845 [s(k)-s(j)]/3, where the pairs (k,j) are given by A205842 and A205843, and s(k) denotes the (k+1)-st Fibonacci number.

1, 2, 1, 4, 6, 11, 7, 18, 14, 7, 29, 28, 27, 47, 41, 77, 76, 75, 48, 125, 124, 123, 96, 48, 203, 199, 192, 185, 328, 322, 281, 532, 528, 521, 514, 329, 861, 857, 850, 843, 658, 329, 1393, 1392, 1391, 1364, 1316, 1268, 2254, 2248, 2207, 1926, 3648

Offset: 1

Author

Clark Kimberling, Feb 01 2012

Keywords

• nonn

Comments

For a guide to related sequences, see A205840.

The first six terms match these differences:

s(4)-s(2) = 5-2 = 3 = 3*1

s(5)-s(2) = 8-2 = 6 = 3*2

s(5)-s(4) = 8-5 = 3 = 3*1

s(6)-s(1) = 13-1 = 12 = 3*4

s(7)-s(3) = 21-3 = 18 = 3*6

s(8)-s(1) = 34-1 = 33 + 3*11

(See the program at A205842.)

Examples

			The first six terms match these differences:
s(4)-s(2) = 5-2 = 3 = 3*1
s(5)-s(2) = 8-2 = 6 = 3*2
s(5)-s(4) = 8-5 = 3 = 3*1
s(6)-s(1) = 13-1 = 12 = 3*4
s(7)-s(3) = 21-3 = 18 = 3*6
s(8)-s(1) = 34-1 = 33 + 3*11
		

Crossrefs

Cf. A204890, A205842, A205845.

Programs

Mathematica

s[n_] := s[n] = Fibonacci[n + 1]; z1 = 400; z2 = 60;
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 = 3; t = d[c]       (* A205841 *)
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}]      (* A205842 *)
Table[j[n], {n, 1, z2}]      (* A205843 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}] (* A205844 *)
Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}] (* A205845 *)