This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A351635 #30 Nov 20 2023 08:15:47
%S A351635 2,6,10,16,38,54,142,196,530,726,1978,2704,7382,10086,27550,37636,
%T A351635 102818,140454,383722,524176,1432070,1956246,5344558,7300804,19946162,
%U A351635 27246966,74440090,101687056,277814198,379501254,1036816702,1416317956,3869452610,5285770566,14440993738,19726764304,53894522342,73621286646
%N A351635 a(n) is the number of perfect matchings of an edge-labeled 2 X n Klein bottle grid graph, or equivalently the number of domino tilings of a 2 X n Klein bottle grid. (The twist is on the length-n side.)
%C A351635 Output of Lu and Wu's formula for the number of perfect matchings of an m X n Klein bottle grid specializes to this sequence for m=2 and the twist on the length-n side.
%H A351635 Sarah-Marie Belcastro, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Belcastro/belcastro4.html">Domino Tilings of 2 X n Grids (or Perfect Matchings of Grid Graphs) on Surfaces</a>, J. Integer Seq. 26 (2023), Article 23.5.6.
%H A351635 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 A351635 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,5,0,-5,0,1).
%F A351635 a(n) = a(n-1) + a(n-2) + (n mod 2)*a(n-1) - 4*(n mod 2).
%F A351635 From _Stefano Spezia_, Feb 15 2022: (Start)
%F A351635 G.f.: 2*x*(1 + 3*x - 7*x^3 - x^4 + 2*x^5)/(1 - 5*x^2 + 5*x^4 - x^6).
%F A351635 a(n) = 5*a(n-2) - 5*a(n-4) + a(n-6) for n > 6. (End)
%e A351635 a(1) = 2 because this is the number of perfect matchings of a 2 X 1 Klein bottle grid graph (one for each choice of the two non-loop edges).
%t A351635 RecurrenceTable[{a[n] ==
%t A351635 a[n - 1] + a[n - 2] + Mod[n, 2] a[n - 1] - 4 Mod[n, 2], a[1] == 2,
%t A351635 a[2] == 6}, a, {n, 1, 50}]
%Y A351635 Cf. A000129, A005248, A001835, A003499, A068397, A103999, A162484, A162483.
%K A351635 nonn,easy
%O A351635 1,1
%A A351635 _Sarah-Marie Belcastro_, Feb 15 2022