A205857 Numbers k for which 6 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.

5, 6, 7, 9, 9, 10, 12, 12, 13, 13, 13, 14, 15, 15, 16, 16, 16, 17, 17, 18, 18, 18, 18, 19, 19, 19, 20, 20, 21, 21, 21, 21, 21, 22, 22, 22, 22, 23, 24, 24, 24, 24, 24, 25, 25, 25, 25, 25, 25, 26, 26, 26, 27, 27, 27, 27, 28, 28, 28, 28

Offset

1,1

Comments

For a guide to related sequences, see A205840.

Links

Table of n, a(n) for n=1..60.

Example

The first six terms match these differences:
s(5)-s(2) = 8-2 = 6 = 6*1
s(6)-s(1) = 13-1 = 12 = 6*2
s(7)-s(3) = 21-3 = 18 = 6*3
s(9)-s(1) = 55-1 = 54 = 6*9
s(9)-s(6) = 55-13 = 42 = 6*7
s(10)-s(4) = 89-5 = 84 =6*14

Mathematica

s[n_] := s[n] = Fibonacci[n + 1]; z1 = 500; 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 = 6; t = d[c]    (* A205856 *)
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}]     (* A205857 *)
Table[j[n], {n, 1, z2}]     (* A205858 *)
Table[s[k[n]]-s[j[n]], {n, 1, z2}]    (* A205859 *)
Table[(s[k[n]]-s[j[n]])/c, {n, 1, z2}]  (* A205860 *)

Crossrefs

Cf. A204892, A205857, A205860.

Sequence in context: A320021 A081407 A268857 * A196026 A191850 A066263

Adjacent sequences: A205854 A205855 A205856 * A205858 A205859 A205860

Keyword

nonn

Author

Clark Kimberling, Feb 02 2012

Status

approved