A274196 -id:A274196 - cp's OEIS frontend

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.

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];

A274201 Limiting reverse row of the array A274196.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 3, 4, 4, 5, 7, 8, 9, 12, 15, 17, 21, 27, 32, 37, 47, 58, 68, 82, 103, 124, 147, 181, 223, 266, 321, 396, 480, 575, 701, 858, 1033, 1248, 1525, 1852, 2232, 2712, 3305, 3998, 4836, 5886, 7148, 8644, 10487, 12752, 15453, 18713, 22731, 27596

Offset: 1

Clark Kimberling, Jun 16 2016

The triangular array (g(n,k)) in A274196 is defined as follows: g(n,k) = 1 for n >= 0; g(n,k) = 0 if k > n; g(n,k) = g(n-1,k-1) + g(n-1,4) for n > 0, k > 1.

Cf. A274196, A274199, A274200.

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]];
z = 300; w = Reverse[Table[g[z, k], {k, 0, z}]];
Take[w, z/3]

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 *)

A274193 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-1,3k) for n > 0, k > 1.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Jun 14 2016

			First  10 rows:
1
1   1
1   1   1
1   1   1   1
1   2   1   1   1
1   2   2   1   1   1
1   2   2   2   1   1   1
1   3   3   2   2   1   1   1
1   3   4   3   2   2   1   1   1
1   4   4   4   3   2   2   1   1   1

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

Cf. A274194 (row sums), A274195, A274200 (limiting reverse row), A274190, 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, 3 k]];
t = Table[g[n, k], {n, 0, 14}, {k, 0, n}]
TableForm[t] (* A274193 array *)
u = Flatten[t] (* A274193 sequence *)

A274198 Decimal expansion of limiting ratio described in Comments.

Original entry on oeis.org

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 *)

cp's OEIS Frontend

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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A274201 Limiting reverse row of the array A274196.

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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.

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A274193 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-1,3k) for n > 0, k > 1.

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A274198 Decimal expansion of limiting ratio described in Comments.

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