A274184 -id:A274184 - cp's OEIS frontend

A274192 Decimal expansion of limiting ratio described in Comments.

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 *)

A274190 Triangular array read by rows: 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.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 3, 3, 2, 1, 1, 1, 4, 4, 3, 2, 1, 1, 1, 5, 6, 5, 3, 2, 1, 1, 1, 7, 8, 7, 5, 3, 2, 1, 1, 1, 9, 12, 10, 8, 5, 3, 2, 1, 1, 1, 13, 17, 15, 11, 8, 5, 3, 2, 1, 1, 1, 18, 24, 22, 17, 12, 8, 5, 3, 2, 1, 1, 1, 25, 35, 32

Offset: 0

Clark Kimberling, Jun 13 2016

			First  10 rows:
1
1   1
1   1   1
1   2   1   1
1   2   2   1   1
1   3   3   2   1   1
1   4   4   3   2   1   1
1   5   6   5   3   2   1   1
1   7   8   7   5   3   2   1   1
1   9   12  10  8   5   3   2   1

Clark Kimberling, Table of n, a(n) for n = 0..10000

Cf. A274184 (row sums), A274192, A274199 (limiting reverse row), A274193, A274196.

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, 14}, {k, 0, n}]
TableForm[t] (* A274190 array *)
u = Flatten[t] (* A274190 sequence *)

A274183 Irregular triangular array having n-th row g(n) defined in Comments.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Jun 13 2016

Let g(0) = (0) and for n > 0, define g(n) inductively as the concatenation of g(n-1) and the numbers k/2 as k ranges through the even numbers k in g(n-1). Every nonnegative integer appears infinitely many times. For the limiting ratio of lengths of consecutive rows, see A274192.

			First six rows:
0
1   0
2   1   0
3   2   1   1   0
4   3   2   2   1   1   0
5   4   3   3   2   2   1   2   1   1   0

Cf. A274184 (row lengths), A274192, A274185.

g[0] = {0}; z = 14; g[n_] := g[n] = Join[g[n - 1] + 1, (1/2) Select[g[n - 1], IntegerQ[#/2] &]]; Flatten[Table[g[n], {n, 0, z}]]

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*)

cp's OEIS Frontend

A274192 Decimal expansion of limiting ratio described in Comments.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A274190 Triangular array read by rows: 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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A274183 Irregular triangular array having n-th row g(n) defined in Comments.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica