A274209 - OEIS

%I #9 Jan 22 2020 07:15:36

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

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

%U 2,9,1,9,3,7,9,7,5,3,2,6,7,7,6,2,7,7

%N Decimal expansion of the reciprocal of the constant in A274192; see Comments.

%C 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 0.67469726387...

%C This constant appears on p. 439 of Tangora's paper cited in Links.

%H 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 Limiting ratio = 0.6746972638734685572768086297549501...

%t z = 1600; g[n_, 0] = g[n, 0] = 1;

%t g[n_, k_] := g[n, k] = If[k > n, 0, g[n - 1, k - 1] + g[n - 1, 2 k]];

%t t = Table[g[n, k], {n, 0, z}, {k, 0, n}];

%t w = Map[Total, t]; (*A274184*)

%t u = N[w[[z]]/w[[z + 1]], 100]

%t d = RealDigits[u][[1]] (*A274209*)

%Y Cf. A274192, A274183, A274184, A274210, A274211.

%K nonn,cons,easy

%O 0,1

%A _Clark Kimberling_, Jun 16 2016