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A263200
Number of perfect matchings on a Möbius strip of width 3 and length 2n.
2
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
2,1
COMMENTS
This sequence obeys the same recurrence relation as A001835.
LINKS
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.
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
MATHEMATICA
CoefficientList[Series[4 (7 - 2 x)/(1 - 4 x + x^2), {x, 0, 33}], x] (* Vincenzo Librandi, Oct 12 2015 *)
PROG
(PARI) Vec(4*x^2*(7-2*x)/(1-4*x+x^2) + O(x^30)) \\ Altug Alkan, Oct 12 2015
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Sergey Perepechko, Oct 12 2015
STATUS
approved