A334135 Number of dimer tilings of a 2*n x 4 Moebius strip.

1, 7, 71, 769, 8449, 93127, 1027207, 11332097, 125019649, 1379271559, 15216755911, 167878292737, 1852111179521, 20433349871431, 225430197438727, 2487050546734081, 27438295728380929, 302712011022644999, 3339659377008916807, 36844671993005504257, 406487518942362537217

Offset

0,2

Links

Table of n, a(n) for n=0..20.

Wikipedia, Chebyshev polynomials

Wikipedia, Resultant

Index entries for linear recurrences with constant coefficients, signature (14,-34,14,-1).

Formula

a(n)^2 = 4^n * Resultant(U(2*n,x), 1+2*x^2+1/2*x^4), where U(n,x) is a Chebyshev polynomial of the second kind and i = sqrt(-1).

G.f.: ((1-x)*(1-6*x+x^2))/(1-14*x+34*x^2-14*x^3+x^4).

a(n) = 14*a(n-1) - 34*a(n-2) + 14*a(n-3) - a(n-4) for n > 3.

Mathematica

a[n_] := 2^n * Sqrt[Resultant[ChebyshevU[2*n, x/2], ChebyshevT[4, I*x/2], x]]; Array[a, 21, 0] (* Amiram Eldar, May 04 2021 *)

Prog

(PARI) a(n) = sqrtint(4^n*polresultant(polchebyshev(2*n, 2, x/2), 1+2*x^2+1/2*x^4))
(PARI) N=20; x='x+O('x^N); Vec(((1-x)*(1-6*x+x^2))/(1-14*x+34*x^2-14*x^3+x^4))

Crossrefs

Column 2 of A103997.

Sequence in context: A048552 A260033 A067307 * A357155 A268702 A363009

Adjacent sequences: A334132 A334133 A334134 * A334136 A334137 A334138

Keyword

nonn,easy

Author

Seiichi Manyama, Apr 15 2020

Status

approved