

A092936


Area of nth triple of hexagons around a triangle.


11



1, 9, 100, 1089, 11881, 129600, 1413721, 15421329, 168220900, 1835008569, 20016873361, 218350598400, 2381839709041, 25981886201049, 283418908502500, 3091626107326449, 33724468272088441, 367877524885646400
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OFFSET

1,2


COMMENTS

This is the unsigned member r=9 of the family of Chebyshev sequences S_r(n) defined in A092184: ((1)^(n+1))*a(n) = S_{9}(n), n>=0.
a(n+1) is the number of tilings of an nboard (a board with dimensions n X 1) using (1/2,1/2)fences, red halfsquares (1/2 X 1 pieces, always placed so that the shorter sides are horizontal), green halfsquares, and blue halfsquares. A (w,g)fence is a tile composed of two w X 1 pieces separated by a gap of width g. a(n+1) also equals the number of tilings of an nboard using (1/4,3/4)fences, red (1/4,1/4)fences, green (1/4,1/4)fences, and blue (1/4,1/4)fences.  Michael A. Allen, Dec 30 2022


LINKS



FORMULA

a(n) = 10*(a(n1)+a(n2))  a(n3).
G.f.: (1x)*x/(110*x10*x^2+x^3).
a(n) = ((3sqrt(13))^n(3+sqrt(13))^n)^2/(13*4^n).
a(n) = 2*(T(n, 11/2)(1)^n)/13 with twice the Chebyshev polynomials of the first kind evaluated at x=11/2: 2*T(n, 11/2)=A057076(n)=((11+3*sqrt(13))^n + (113*sqrt(13))^n)/2^n.  Wolfdieter Lang, Oct 18 2004
a(n+1) = 11*a(n)  a(n1) + 2*(1)^n.
a(n+1) = (1 + (1)^n)/2 + 9*Sum_{k=1..n} ( k*a(n+1k) ). (End)


EXAMPLE

a(5) = 10*(1089+100)9 = 11881. From A006190, a(5) = (3*33+10)^2 = 11881.


MAPLE



MATHEMATICA

CoefficientList[Series[(1x)*x/(110*x10*x^2+x^3), {x, 0, 20}], x]
(CoefficientList[Series[x/(13*xx^2), {x, 0, 20}], x])^2
Table[Round[((3+Sqrt[13])^n)^2/(13*4^n)], {n, 0, 20}]
LinearRecurrence[{10, 10, 1}, {1, 9, 100}, 18] (* Georg Fischer, Feb 22 2019 *)


PROG

(GAP) a:=[1, 9, 100];; for n in [4..18] do a[n]:=10*(a[n1]+a[n2])a[n3]; od; a; # Muniru A Asiru, Feb 20 2018


CROSSREFS



KEYWORD

easy,nonn


AUTHOR



STATUS

approved



