A369359 a(n) is the total semiperimeter over all Motzkin polyominoes of length n.

0, 2, 4, 11, 29, 80, 222, 624, 1766, 5030, 14396, 41371, 119297, 345008, 1000274, 2906427, 8461269, 24674718, 72065892, 210766089, 617173791, 1809257448, 5309289426, 15594735954, 45845032212, 134880781266, 397123496252, 1170026790029, 3449372893511, 10175133060424

Offset

0,2

Links

Table of n, a(n) for n=0..29.

Jean-Luc Baril, Sergey Kirgizov, José L. Ramírez, and Diego Villamizar, The Combinatorics of Motzkin Polyominoes, arXiv:2401.06228 [math.CO], 2024. See Corollary 4.4 and Corollary 4.5 at pages 9-10.

Formula

From Corollary 4.4 in Baril et al.: (Start)

G.f.: (1 + x^2 - (1 + x)*sqrt(1 - 2*x - 3*x^2))/(2*x*sqrt(1 - 2*x - 3*x^2)).

a(n) ~ (5/6)*sqrt(3/Pi)*3^n/sqrt(n). (End)

a(n) = A002426(n) + 2*A002426(n-1) - 2*A001006(n-1) for n > 0. [Corollary 4.5 in Baril et al.]

Maple

gf := ((x^2+1)/sqrt((1-3*x)*(x+1))-(x+1))/(2*x): ser := series(gf, x, 40):
seq(coeff(ser, x, k), k = 0..29);  # Peter Luschny, Jan 21 2024

Mathematica

a[n_]:=SeriesCoefficient[(1+x^2-(1+x)Sqrt[1-2x-3x^2])/(2x*Sqrt[1-2x-3x^2]), {x, 0, n}]; Array[a, 30, 0]

Crossrefs

Cf. A001006, A002426, A369360.

Sequence in context: A132836 A148140 A328139 * A148141 A317879 A148142

Adjacent sequences: A369356 A369357 A369358 * A369360 A369361 A369362

Keyword

nonn

Author

Stefano Spezia, Jan 21 2024

Status

approved