A274199 Limiting reverse row of the array A274190.

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

0,3

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)

Links

Table of n, a(n) for n=0..41.

Daniel Gabric and Jeffrey Shallit, Smallest and Largest Block Palindrome Factorizations, arXiv:2302.13147 [math.CO], 2023.

Example

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.

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 *)

Crossrefs

Cf. A274190, A274200, A274201.

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

Sequence in context: A355975 A327421 A124062 * A099823 A358335 A240523

Adjacent sequences: A274196 A274197 A274198 * A274200 A274201 A274202

Keyword

nonn,easy

Author

Clark Kimberling, Jun 13 2016

Status

approved