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

%I A281112 #12 Nov 20 2024 04:18:26
%S A281112 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 A281112 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 A281112 9,2,9,4,3,8,6,0,4,8,4,5,9,7,5,6,9,4
%N A281112 Decimal expansion of (conjectured) limit c(n+1)/c(n), where c = A078140.
%H A281112 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 A281112 Limit_{n->oo} c(n+1)/c(n) = 1.688924110769165206686359... (conjectured)
%t A281112 z = 3880; r = GoldenRatio;
%t A281112 f[x_] := f[x] = Sum[Floor[r*(k + 1)] x^k, {k, 0, z}];
%t A281112 t = CoefficientList[Series[1/f[x], {x, 0, z}], x];(* A078140 *)
%t A281112 Table[N[t[[n]]/t[[n - 1]], 80], {n, 2, z, 100}]
%t A281112 u = N[t[[z]]/t[[z - 1]], 120]
%t A281112 RealDigits[Abs[u], 10][[1]] (* A281112 *)
%Y A281112 Cf. A078140, A001622.
%K A281112 nonn,cons
%O A281112 1,2
%A A281112 _Clark Kimberling_, Feb 10 2017