cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A274200 Limiting reverse row of the array A274193.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 20, 27, 35, 45, 60, 77, 101, 132, 170, 223, 289, 375, 490, 634, 826, 1074, 1392, 1813, 2352, 3055, 3973, 5154, 6700, 8702, 11296, 14681, 19058, 24754, 32156, 41747, 54232, 70425, 91451, 118786, 154241, 200319, 260157
Offset: 1

Views

Author

Clark Kimberling, Jun 16 2016

Keywords

Comments

The triangular array (g(n,k)) at A274193 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-1,3) for n > 0, k > 1.

Examples

			Row g(9,k):  1,4,4,4,3,2,2,1,1,1; the reversal is
1,1,1,2,2,3,4,4,4,1,..., which agrees with A274200 up to the first 7 terms.

Crossrefs

Cf. A274193, A274199, A274201.

Programs

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

A274194 Row sums of the array A274193, 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-1,3k) for n > 0, k > 1.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 14, 18, 23, 31, 40, 52, 68, 88, 114, 148, 192, 249, 323, 420, 545, 707, 919, 1191, 1547, 2009, 2606, 3386, 4396, 5707, 7413, 9623, 12496, 16228, 21069, 27363, 35531, 46139, 59920, 77806, 101043, 131215, 170392, 221283, 287355, 373168
Offset: 0

Views

Author

Clark Kimberling, Jun 14 2016

Keywords

Comments

See A274195 for the limit of a(n)/a(n-1).

Examples

			(g(9,k)) = (1,4,4,4,3,2,2,1,1,1), so that a(9) = 23.

Crossrefs

Cf. A274193, A274195.

Programs

  • Mathematica
    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, 3 k]];
t = Table[g[n, k], {n, 0, z}, {k, 0, n}]; w = Map[Total, t];

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

Views

Author

Clark Kimberling, Jun 13 2016

Keywords

Examples

			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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

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

Views

Author

Clark Kimberling, Jun 16 2016

Keywords

Comments

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).

Examples

			Limiting ratio = 1.2986406408617046456933441615...

Crossrefs

Cf. A274193, A274194, A274198, A274210 (reciprocal).

Programs

  • Mathematica
    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 *)

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

Views

Author

Clark Kimberling, Jun 16 2016

Keywords

Comments

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).

Examples

			limiting ratio = 1.21297992704936771891526402555...

Crossrefs

Cf. A274196, A274197, A274192, A274193, A274211 (reciprocal).

Programs

  • Mathematica
    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 *)
Showing 1-5 of 5 results.

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