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a(n) is the total semiperimeter over all Motzkin polyominoes of length n.
1

%I #14 Jan 22 2024 01:52:59

%S 0,2,4,11,29,80,222,624,1766,5030,14396,41371,119297,345008,1000274,

%T 2906427,8461269,24674718,72065892,210766089,617173791,1809257448,

%U 5309289426,15594735954,45845032212,134880781266,397123496252,1170026790029,3449372893511,10175133060424

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

%H Jean-Luc Baril, Sergey Kirgizov, José L. Ramírez, and Diego Villamizar, <a href="https://arxiv.org/abs/2401.06228">The Combinatorics of Motzkin Polyominoes</a>, arXiv:2401.06228 [math.CO], 2024. See Corollary 4.4 and Corollary 4.5 at pages 9-10.

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

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

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

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

%p gf := ((x^2+1)/sqrt((1-3*x)*(x+1))-(x+1))/(2*x): ser := series(gf, x, 40):

%p seq(coeff(ser, x, k), k = 0..29); # _Peter Luschny_, Jan 21 2024

%t 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]

%Y Cf. A001006, A002426, A369360.

%K nonn

%O 0,2

%A _Stefano Spezia_, Jan 21 2024