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.)

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

Author

Sarah-Marie Belcastro, Feb 15 2022

Keywords

• nonn
• easy

Comments

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.

Examples

			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).
		

Links

• 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).

Crossrefs

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

Programs

Mathematica

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}]

Formula

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)