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A369359
a(n) is the total semiperimeter over all Motzkin polyominoes of length n.
1
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
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
KEYWORD
nonn
AUTHOR
Stefano Spezia, Jan 21 2024
STATUS
approved