A274197 -id:A274197 - cp's OEIS frontend

A274196 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-1,4k) for n > 0, k > 1.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 3, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 3, 4, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 4, 4, 4, 4, 3, 2, 2

Offset: 0

Clark Kimberling, Jun 16 2016

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

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

Cf. A274197 (row sums), A274201 (limiting reverse row), A274190.

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, 14}, {k, 0, n}]
TableForm[t] (* A274196 array *)
u = Flatten[t] (* A274196 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 *)

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

Original entry on oeis.org

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*)

cp's OEIS Frontend

A274196 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-1,4k) for n > 0, k > 1.

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A274198 Decimal expansion of limiting ratio described in Comments.

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

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