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A274192 Decimal expansion of limiting ratio described in Comments.

%I A274192 #9 Jan 22 2020 14:37:46
%S A274192 1,4,8,2,1,4,6,2,2,1,0,4,5,7,9,6,4,7,3,9,5,1,0,9,4,5,0,5,0,8,9,2,9,2,
%T A274192 1,8,8,1,0,0,7,2,2,0,9,9,2,0,0,8,2,7,9,6,3,7,8,9,8,7,8,3,7,8,8,6,2,3,
%U A274192 2,4,8,7,2,9,2,5,0,1,6,9,3,4,8,5,8,6
%N A274192 Decimal expansion of limiting ratio described in Comments.
%C A274192 As in A274190, define g(n,k) = 1 for n >= 0; g(n,k) = 0 if k > n; g(n,k) = g(n-1,k-1) + g(n-1,2k) for n > 0, k > 1.  The sum of numbers in the n-th row of the array {g(n,k)} is given by A274184; viz., this sum is also the number of numbers in the n-th row of the array in A274183. In other words, if we put h(0) = (0) and for n > 0 define h(n) inductively as the concatenation of h(n-1) and the numbers k/2 as k ranges through the even numbers k in h(n-1), and then let H(n) be the number of numbers in h(n), then H(n)/H(n-1) approaches 1.48214622...
%C A274192 This constant appears on p. 439 of Tangora's paper cited in Links.
%H A274192 M. C. Tangora, <a href="https://doi.org/10.1016/S0195-6698(13)80019-4">Level number sequences of trees and the lambda algebra</a>, European J. Combinatorics 12 (1991), 433-443.
%e A274192 Limiting ratio = 1.48214622104579647395109450508929...
%t A274192 z = 1500; g[n_, 0] = g[n, 0] = 1;
%t A274192 g[n_, k_] := g[n, k] = If[k > n, 0, g[n - 1, k - 1] + g[n - 1, 2 k]];
%t A274192 t = Table[g[n, k], {n, 0, z}, {k, 0, n}];
%t A274192 w = Map[Total, t];   (* A274184 *)
%t A274192 u = N[w[[z]]/w[[z - 1]], 100]
%t A274192 RealDigits[u][[1]] (* A274192 *)
%Y A274192 Cf. A274190, A274184, A274195, A274198, A274209 (reciprocal).
%K A274192 nonn,cons,easy
%O A274192 1,2
%A A274192 _Clark Kimberling_, Jun 13 2016