

A274199


Limiting reverse row of the array A274190.


22



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
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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(n1,k1) + g(n1,2k) for n > 0, k > 1.
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



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*#[[i1]], {i, 2, Length[#]}]&]], {n, 15}] (* Gus Wiseman, Mar 12 2021 *)


CROSSREFS

Cf. A000929, A003242, A154402, A224957, A342094, A342095, A342096, A342097, A342098, A342191, A342330A342342.


KEYWORD

nonn,easy


AUTHOR



STATUS

approved



