A068397 -id:A068397 - cp's OEIS frontend

A324482 Symmetric inflation orbit counts (b-bar)_{2n} for 1D cut and project patterns with inversion symmetric tau-inflation.

Original entry on oeis.org

0, 4, 0, 8, 10, 12, 28, 48, 72, 120, 198, 312, 520, 840, 1350

Offset: 1

N. J. A. Sloane, Mar 12 2019

a(n)/(2*n) is probably in the OEIS, but at present there not enough terms to identify it uniquely.

M. Baake, J. Hermisson, P. Pleasants, The torus parametrization of quasiperiodic LI-classes, J. Phys. A 30 (1997), no. 9, 3029-3056. See Table 3.

Cf. A068397.

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.)

Original entry on oeis.org

2, 6, 10, 16, 38, 54, 142, 196, 530, 726, 1978, 2704, 7382, 10086, 27550, 37636, 102818, 140454, 383722, 524176, 1432070, 1956246, 5344558, 7300804, 19946162, 27246966, 74440090, 101687056, 277814198, 379501254, 1036816702, 1416317956, 3869452610, 5285770566, 14440993738, 19726764304, 53894522342, 73621286646

Offset: 1

Sarah-Marie Belcastro, Feb 15 2022

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.

			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).

Sarah-Marie Belcastro, Domino Tilings of 2 X n Grids (or Perfect Matchings of Grid Graphs) on Surfaces, J. Integer Seq. 26 (2023), Article 23.5.6.
W. T. Lu and F. Y. Wu, Close-packed dimers on nonorientable surfaces, Physics Letters A, 293 (2002), 235-246.
Index entries for linear recurrences with constant coefficients, signature (0,5,0,-5,0,1).

Cf. A000129, A005248, A001835, A003499, A068397, A103999, A162484, A162483.

RecurrenceTable[{a[n] ==
   a[n - 1] + a[n - 2] + Mod[n, 2] a[n - 1] - 4 Mod[n, 2], a[1] == 2,
  a[2] == 6}, a, {n, 1, 50}]

a(n) = a(n-1) + a(n-2) + (n mod 2)*a(n-1) - 4*(n mod 2).
From Stefano Spezia, Feb 15 2022: (Start)
G.f.: 2*x*(1 + 3*x - 7*x^3 - x^4 + 2*x^5)/(1 - 5*x^2 + 5*x^4 - x^6).
a(n) = 5*a(n-2) - 5*a(n-4) + a(n-6) for n > 6. (End)

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A324482 Symmetric inflation orbit counts (b-bar)_{2n} for 1D cut and project patterns with inversion symmetric tau-inflation.

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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.)

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