%I #9 Jul 10 2017 22:50:08

%S 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,

%T 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,

%U 9,2,9,4,3,8,6,0,4,8,4,5,9,7,5,6,9,4

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

%H Clark Kimberling, <a href="http://mathoverflow.net/questions/259821/another-question-about-the-golden-ratio-and-other-numbers"> Another question about the golden ratio and other numbers</a>, MathOverflow, Jan 17 2017.

%e c(n+1)/c(n)) -> 1.688924110769165206686359... (conjectured)

%t z = 3880; r = GoldenRatio;

%t f[x_] := f[x] = Sum[Floor[r*(k + 1)] x^k, {k, 0, z}];

%t t = CoefficientList[Series[1/f[x], {x, 0, z}], x];(* A078140 *)

%t Table[N[t[[n]]/t[[n - 1]], 80], {n, 2, z, 100}]

%t u = N[t[[z]]/t[[z - 1]], 120]

%t RealDigits[Abs[u], 10][[1]] (* A281112 *)

%Y Cf. A078140, A001622.

%K nonn,cons

%O 1,2

%A _Clark Kimberling_, Feb 10 2017