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 1-3 of 3 results.

A274199 Limiting reverse row of the array A274190.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 12, 19, 29, 44, 67, 101, 152, 228, 342, 511, 763, 1138, 1695, 2523, 3752, 5578, 8287, 12307, 18272, 27119, 40241, 59700, 88556, 131340, 194772, 288815, 428229, 634900, 941263, 1395397, 2068560, 3066372, 4545387, 6737633, 9987026, 14803303
Offset: 0

Views

Author

Clark Kimberling, Jun 13 2016

Keywords

Comments

The triangular array (g(n,k)) at A274190 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-1,2k) for n > 0, k > 1.
From Gus Wiseman, Mar 12 2021: (Start)
Also (apparently) the number of compositions of n where all adjacent parts (x, y), satisfy x < 2y. For example, the a(1) = 1 through a(6) = 12 compositions are:
(1) (2) (3) (4) (5) (6)
(11) (12) (13) (14) (15)
(111) (22) (23) (24)
(112) (32) (33)
(1111) (113) (114)
(122) (123)
(1112) (132)
(11111) (222)
(1113)
(1122)
(11112)
(111111)
(End)

Examples

			Row (g(14,k)):  1, 51, 73, 69, 55, 40, 28, 19, 12, 8, 5, 3, 2, 1, 1; the reversal is 1 1 2 3 5 8 12 19 28 ..., which agrees with A274199 up to 19.

Links

Crossrefs

Cf. A274190, A274200, A274201.
Cf. A000929, A003242, A154402, A224957, A342094, A342095, A342096, A342097, A342098, A342191, A342330-A342342.

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]];
z = 300; u = Reverse[Table[g[z, k], {k, 0, z}]];
z = 301; v = Reverse[Table[g[z, k], {k, 0, z}]];
w = Join[{1}, Intersection[u, v]] (* A274199 *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],And@@Table[#[[i]]<2*#[[i-1]],{i,2,Length[#]}]&]],{n,15}] (* Gus Wiseman, Mar 12 2021 *)

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.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jun 16 2016

Keywords

Examples

			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

Links

Crossrefs

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

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, 4 k]];
t = Table[g[n, k], {n, 0, 14}, {k, 0, n}]
TableForm[t] (* A274196 array *)
u = Flatten[t] (* A274196 sequence *)

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]
Showing 1-3 of 3 results.

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