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A369360
a(n) is the number of interior points over all Motzkin polyominoes of length n.
1
1, 6, 25, 93, 324, 1088, 3565, 11487, 36564, 115327, 361194, 1124932, 3487857, 10774350, 33181365, 101924727, 312402600, 955728009, 2919095808, 8903220174, 27121002621, 82525328253, 250867833102, 761946207338, 2312406692839, 7012910929818, 21254741084817, 64381661704933
OFFSET
3,2
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 6.4 at pages 18-19.
FORMULA
a(n) = (3^n - 3*A002426(n))/2 - 2*A002426(n-1) + 2*A001006(n-1).
MATHEMATICA
t[n_]:=SeriesCoefficient[1/(Sqrt[1-2x-3x^2]), {x, 0, n}]; (* A002426 *)
m[n_]:=SeriesCoefficient[(1 - x - (1 - 2x - 3x^2)^(1/2))/(2x^2), {x, 0, n}]; (* A001006 *)
a[n_]:=(3^n-3t[n])/2-2t[n-1]+2m[n-1]; Array[a, 28, 3]
CROSSREFS
KEYWORD
nonn
AUTHOR
Stefano Spezia, Jan 21 2024
STATUS
approved