A281112 Decimal expansion of (conjectured) limit c(n+1)/c(n), where c = A078140.

1, 6, 8, 8, 9, 2, 4, 1, 1, 0, 7, 6, 9, 1, 6, 5, 2, 0, 6, 6, 8, 6, 3, 5, 9, 6, 4, 3, 7, 1, 9, 8, 3, 3, 6, 0, 8, 9, 6, 1, 4, 6, 2, 6, 4, 6, 6, 1, 6, 6, 7, 2, 1, 9, 1, 6, 4, 5, 6, 6, 3, 5, 6, 6, 6, 4, 0, 8, 9, 2, 9, 4, 3, 8, 6, 0, 4, 8, 4, 5, 9, 7, 5, 6, 9, 4

Offset: 1

Author

Clark Kimberling, Feb 10 2017

Keywords

• nonn
• cons

Examples

			Limit_{n->oo} c(n+1)/c(n) = 1.688924110769165206686359... (conjectured)
		

Links

• Clark Kimberling, Another question about the golden ratio and other numbers, MathOverflow, Jan 17 2017.

Crossrefs

Cf. A078140, A001622.

Programs

Mathematica

z = 3880; r = GoldenRatio;
f[x_] := f[x] = Sum[Floor[r*(k + 1)] x^k, {k, 0, z}];
t = CoefficientList[Series[1/f[x], {x, 0, z}], x];(* A078140 *)
Table[N[t[[n]]/t[[n - 1]], 80], {n, 2, z, 100}]
u = N[t[[z]]/t[[z - 1]], 120]
RealDigits[Abs[u], 10][[1]] (* A281112 *)