A263200 Number of perfect matchings on a Möbius strip of width 3 and length 2n.
28, 104, 388, 1448, 5404, 20168, 75268, 280904, 1048348, 3912488, 14601604, 54493928, 203374108, 759002504, 2832635908, 10571541128, 39453528604, 147242573288, 549516764548, 2050824484904, 7653781175068, 28564300215368, 106603419686404, 397849378530248
Offset: 2
Links
- Colin Barker, Table of n, a(n) for n = 2..1000
- W. T. Lu and F. Y. Wu, Close-packed dimers on nonorientable surfaces, Physics Letters A, 293(2002), 235-246.
- S. N. Perepechko, Recurrence relations for the number of perfect matchings on the Mobius strips (in Russian), Proc. of XIX international conference on computational mechanics and modern applied software systems (CMMASS'2015), Alushta, Crimea, 2015, 98-100.
- Sergey Perepechko, Graph view
- G. Tesler, Matchings in graphs on non-orientable surfaces, Journal of Combinatorial Theory B, 78(2000), 198-231.
- Index entries for linear recurrences with constant coefficients, signature (4,-1).
Programs
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Magma
I:=[28,104]; [n le 2 select I[n] else 4*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Oct 12 2015
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Mathematica
CoefficientList[Series[4 (7 - 2 x)/(1 - 4 x + x^2), {x, 0, 33}], x] (* Vincenzo Librandi, Oct 12 2015 *) -
PARI
Vec(4*x^2*(7-2*x)/(1-4*x+x^2) + O(x^30)) \\ Altug Alkan, Oct 12 2015
Formula
a(n) = Product_{k=1..n} (10 + 2*cos(Pi*(4*k-1)/n) - 12*cos(1/2*Pi*(4*k-1)/n)).
G.f.: 4*x^2*(7-2*x)/(1-4*x+x^2).
From Colin Barker, Oct 12 2015: (Start)
a(n) = 2*((2-sqrt(3))^n + (2+sqrt(3))^n).
a(n) = 4*a(n-1) - a(n-2). (End)
a(n) = 4*A001075(n) for n >= 2. - Philippe Deléham, Mar 03 2023
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