A274194 -id:A274194 - cp's OEIS frontend

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.

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 *)

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 *)

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*)

cp's OEIS Frontend

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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A274195 Decimal expansion of limiting ratio described in Comments.

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

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