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%I #32 Mar 04 2023 00:01:05
%S 28,104,388,1448,5404,20168,75268,280904,1048348,3912488,14601604,
%T 54493928,203374108,759002504,2832635908,10571541128,39453528604,
%U 147242573288,549516764548,2050824484904,7653781175068,28564300215368,106603419686404,397849378530248
%N Number of perfect matchings on a Möbius strip of width 3 and length 2n.
%C This sequence obeys the same recurrence relation as A001835.
%H Colin Barker, <a href="/A263200/b263200.txt">Table of n, a(n) for n = 2..1000</a>
%H W. T. Lu and F. Y. Wu, <a href="http://dx.doi.org/10.1016/S0375-9601(02)00019-1">Close-packed dimers on nonorientable surfaces</a>, Physics Letters A, 293(2002), 235-246.
%H S. N. Perepechko, <a href="http://www.cmmass.ru/files/cmmass2015_web.pdf">Recurrence relations for the number of perfect matchings on the Mobius strips (in Russian)</a>, Proc. of XIX international conference on computational mechanics and modern applied software systems (CMMASS'2015), Alushta, Crimea, 2015, 98-100.
%H Sergey Perepechko, <a href="/A263200/a263200.png">Graph view</a>
%H G. Tesler, <a href="http://dx.doi.org/10.1006/jctb.1999.1941">Matchings in graphs on non-orientable surfaces</a>, Journal of Combinatorial Theory B, 78(2000), 198-231.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-1).
%F a(n) = Product_{k=1..n} (10 + 2*cos(Pi*(4*k-1)/n) - 12*cos(1/2*Pi*(4*k-1)/n)).
%F G.f.: 4*x^2*(7-2*x)/(1-4*x+x^2).
%F From _Colin Barker_, Oct 12 2015: (Start)
%F a(n) = 2*((2-sqrt(3))^n + (2+sqrt(3))^n).
%F a(n) = 4*a(n-1) - a(n-2). (End)
%F a(n) = 4*A001075(n) for n >= 2. - _Philippe Deléham_, Mar 03 2023
%t CoefficientList[Series[4 (7 - 2 x)/(1 - 4 x + x^2), {x, 0, 33}], x] (* _Vincenzo Librandi_, Oct 12 2015 *)
%o (PARI) Vec(4*x^2*(7-2*x)/(1-4*x+x^2) + O(x^30)) \\ _Altug Alkan_, Oct 12 2015
%o (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
%Y Cf. A001075, A001835, A020878.
%K nonn,easy
%O 2,1
%A _Sergey Perepechko_, Oct 12 2015
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