A274198 -id:A274198 - cp's OEIS frontend

A274211 Decimal expansion of the reciprocal of the constant in A274198.

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

Offset: 0

Clark Kimberling, Jun 16 2016

			limiting ratio = 0.82441595091564978969840454...

Cf. A274198, A274197, A274209, A274210.

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, 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]] (*A274211*)

A274192 Decimal expansion of limiting ratio described in Comments.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jun 13 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 1.48214622...

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

			Limiting ratio = 1.48214622104579647395109450508929...

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

Cf. A274190, A274184, A274195, A274198, A274209 (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, 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]
RealDigits[u][[1]] (* A274192 *)

A274195 Decimal expansion of limiting ratio described in Comments.

Original entry on oeis.org

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

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-1,3k) for n > 0, k > 1. The sum of numbers in the n-th row of the array {g(n,k)} is given by A274194, and "limiting ratio" = limit of A274194(n)/A274194(n-1).

			Limiting ratio = 1.2986406408617046456933441615...

Cf. A274193, A274194, A274198, A274210 (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, 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]] (* A274195 *)

A274197 Row sums of the array A274196, defined by 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.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 13, 17, 21, 25, 30, 38, 46, 55, 67, 83, 100, 121, 148, 180, 217, 264, 321, 388, 470, 572, 693, 838, 1018, 1237, 1497, 1814, 2205, 2675, 3239, 3930, 4773, 5785, 7010, 8510, 10326, 12514, 15177, 18422, 22340, 27084, 32862, 39872, 483458

Offset: 0

Clark Kimberling, Jun 16 2016

See A274198 for the limit of a(n)/a(n-1).

			(g(7,k)) = (1,2,2,2,1,1,1,1), so that a(7) = 11.

Cf. A274196, A274198.

z = 150; 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];

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A274211 Decimal expansion of the reciprocal of the constant in A274198.

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A274192 Decimal expansion of limiting ratio described in Comments.

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A274195 Decimal expansion of limiting ratio described in Comments.

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A274197 Row sums of the array A274196, defined by 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.

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