A274211 -id:A274211 - cp's OEIS frontend

A274198 Decimal expansion of limiting ratio described in Comments.

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

Offset: 1

Clark Kimberling, Jun 16 2016

As in A274193, 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-1,4k) for n > 0, k > 1. The sum of numbers in the n-th row of the array {g(n,k)} is given by A274197, and "limiting ratio" = limit of A274197(n)/A274197(n-1).

			limiting ratio = 1.21297992704936771891526402555...

Cf. A274196, A274197, A274192, A274193, A274211 (reciprocal).

z = 1500; g[n_, 0] = g[n, 0] = 1;
g[n_, k_] := g[n, k] = If[k > n, 0, g[n - 1, k - 1] + g[n - 1, 4 k]];
t = Table[g[n, k], {n, 0, z}, {k, 0, n}];
w = Map[Total, t];   (* A274197 *)
u = N[w[[z]]/w[[z - 1]], 100]
RealDigits[u][[1]] (* A274198 *)

A274209 Decimal expansion of the reciprocal of the constant in A274192; see Comments.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Jun 16 2016

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...

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

			Limiting ratio = 0.6746972638734685572768086297549501...

M. C. Tangora, Level number sequences of trees and the lambda algebra, European J. Combinatorics 12 (1991), 433-443.

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

z = 1600; g[n_, 0] = g[n, 0] = 1;
g[n_, k_] := g[n, k] = If[k > n, 0, g[n - 1, k - 1] + g[n - 1, 2 k]];
t = Table[g[n, k], {n, 0, z}, {k, 0, n}];
w = Map[Total, t]; (*A274184*)
u = N[w[[z]]/w[[z + 1]], 100]
d = RealDigits[u][[1]] (*A274209*)

A274210 Decimal expansion of the reciprocal of the constant in A274195.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Jun 16 2016

			Limiting ratio = 0.770035965712929198792214166196...

Cf. A274195, A274194, A274195, A274209, A274211.

z = 1600; g[n_, 0] = g[n, 0] = 1;
g[n_, k_] := g[n, k] = If[k > n, 0, g[n - 1, k - 1] + g[n - 1, 3 k]];
t = Table[g[n, k], {n, 0, z}, {k, 0, n}];
w = Map[Total, t]; (*A274194*)
u = N[w[[z]]/w[[z + 1]], 100]
RealDigits[u][[1]] (*A274210*)

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A274198 Decimal expansion of limiting ratio described in Comments.

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A274209 Decimal expansion of the reciprocal of the constant in A274192; see Comments.

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A274210 Decimal expansion of the reciprocal of the constant in A274195.

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