OFFSET
1,1
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.
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).
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)
EXAMPLE
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).
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}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Sarah-Marie Belcastro, Feb 15 2022
STATUS
approved