Sarah-Marie Belcastro - cp's OEIS frontend

User: Sarah-Marie Belcastro

Sarah-Marie Belcastro has authored 4 sequences.

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

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)

A162483 a(n) is the number of perfect matchings of an edge-labeled 2 X (2n+1) Mobius grid graph.

Original entry on oeis.org

3, 6, 13, 31, 78, 201, 523, 1366, 3573, 9351, 24478, 64081, 167763, 439206, 1149853, 3010351, 7881198, 20633241, 54018523, 141422326, 370248453, 969323031, 2537720638, 6643838881, 17393796003, 45537549126, 119218851373, 312119004991, 817138163598

Offset: 0

Sarah-Marie Belcastro, Jul 04 2009

This is a specialization for m=2 of a general formula for the number of perfect matchings of an edge-labeled m X (2n+1) Mobius grid graph.

			G.f. = 3 + 6*x + 13*x^2 + 31*x^3 + 78*x^4 + 201*x^5 + 523*x^6 + 1366*x^7 + ...
a(0) = 3 because this is the number of perfect matchings of a 2 X 1 Mobius grid graph (one for each of the three multiple edges).

Clark Kimberling, Table of n, a(n) for n = 0..1000
G. Tesler, Matchings in graphs on non-orientable surfaces, Journal of Combinatorial Theory B, 78(2000), 198-231.
Index entries for linear recurrences with constant coefficients, signature (4,-4,1).

Cf. A020878, A256233.

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((3-6*x+x^2)/((1-x)*(x^2-3*x+1)))); // G. C. Greubel, Sep 22 2018

Table[Re[(1 - I) (2*I + Fibonacci[2 + 2*n] + 1/2 (-Fibonacci[1 + 2*n] + LucasL[1 + 2*n]))], {n, 0, 30}]
Table[LucasL[2*n + 1] + 2, {n, 0, 30}] (* Clark Kimberling, Oct 26 2012 *)
LinearRecurrence[{4, -4, 1}, {3, 6, 13}, 30] (* or *) CoefficientList[Series[(-3 + 6 x - x^2)/(-1 + 4 x - 4 x^2 + x^3), {x, 0, 30}], x] (* Stefano Spezia, Sep 23 2018 *)

{a(n) = 2 + fibonacci(2*n) + fibonacci(2*n+2)}; /* Michael Somos, Nov 03 2016 */

a(n) = Real((1-I) * ((L(2*n+1) - F(2*n+1))/2 + F(2*n+2) + 2*I)).
From R. J. Mathar, Aug 08 2009: (Start)
a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3).
G.f.: (3-6*x+x^2)/((1-x)*(x^2-3*x+1)). (End)
a(n+1)-a(n) = A005248(n+1). - R. J. Mathar, Dec 18 2010
a(n) = A000032(2n+1)+2. - Clark Kimberling, Oct 26 2012
a(n) = 2^(-1-n)*(2^(2+n)-(3-sqrt(5))^n*(-1+sqrt(5))+(1+sqrt(5))*(3+sqrt(5))^n). - Colin Barker, Nov 03 2016
a(n) = 2 + L(2*n+1), A256233(n) = -a(-n-1) for all n in Z. - Michael Somos, Nov 03 2016

A162484 a(1) = 2, a(2) = 8; a(n) = 2 a(n - 1) + a(n - 2) - 4*(n mod 2).

Original entry on oeis.org

2, 8, 14, 36, 82, 200, 478, 1156, 2786, 6728, 16238, 39204, 94642, 228488, 551614, 1331716, 3215042, 7761800, 18738638, 45239076, 109216786, 263672648, 636562078, 1536796804, 3710155682, 8957108168, 21624372014, 52205852196, 126036076402, 304278005000

Offset: 1

Sarah-Marie Belcastro, Jul 04 2009

a(n) is the number of perfect matchings of an edge-labeled 2 X n toroidal grid graph, or equivalently the number of domino tilings of a 2 X n toroidal grid.

			a(3) = 2 a(2) + a(1) - 4*(3 mod 2) = 2*8 + 2 - 4 = 14.

Colin Barker, Table of n, a(n) for n = 1..1000
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.
Index entries for linear recurrences with constant coefficients, signature (2,2,-2,-1).

Cf. A000129.

Fold[Append[#1, 2 #1[[#2 - 1]] + #1[[#2 - 2]] - 4 Mod[#2, 2]] &, {2, 8}, Range[3, 30]] (* or *)
Rest@ CoefficientList[Series[-2 x (-1 - 2 x + 3 x^2 + 2 x^3)/((x - 1) (1 + x) (x^2 + 2 x - 1)), {x, 0, 30}], x] (* Michael De Vlieger, Dec 16 2017 *)
LinearRecurrence[{2,2,-2,-1},{2,8,14,36},30] (* Harvey P. Dale, Aug 24 2018 *)

for n > 2, (1/2) ((1 + sqrt(2))^n (2 - (-2 + sqrt(2)) (-1 + sqrt(2))^(2 floor(n/2))) + (1 - sqrt(2))^n (2 + (1 + sqrt(2))^(2 floor(n/2)) (2 + sqrt(2)))) (from Mathematica's solution to the recurrence).
Pell(n) + Pell(n-2) + 2*((n-1) mod 2).
From R. J. Mathar, Jul 26 2009: (Start)
a(n)= 2*a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) = 2*A100828(n-1).
G.f.: -2*x*(-1-2*x+3*x^2+2*x^3)/((x-1)*(1+x)*(x^2+2*x-1)).
(End)
a(n) = 1 + (-1)^n + (1-sqrt(2))^n + (1+sqrt(2))^n. - Colin Barker, Dec 16 2017

A162485 a(1)=4, a(2)=6; for n > 2, a(n) = 2a(n-1) + a(n-2) - 4((n-1) mod 2).

Original entry on oeis.org

4, 6, 16, 34, 84, 198, 480, 1154, 2788, 6726, 16240, 39202, 94644, 228486, 551616, 1331714, 3215044, 7761798, 18738640, 45239074, 109216788, 263672646, 636562080, 1536796802, 3710155684, 8957108166, 21624372016, 52205852194, 126036076404, 304278004998, 734592086400

Offset: 1

Sarah-Marie Belcastro, Jul 04 2009

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-2 side.)

			a(3) = 2*a(2) + a(1) - 4*(2 mod 2) = 2*6 + 4 - 0 = 16.

Paolo Xausa, Table of n, a(n) for n = 1..1000
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.
Index entries for linear recurrences with constant coefficients, signature (2,2,-2,-1).

Cf. A002203.

LinearRecurrence[{2, 2, -2, -1}, {4, 6, 16, 34}, 50] (* Paolo Xausa, Jun 27 2025 *)

For n > 1, a(n) = (1/2)*((1 + sqrt(2))^n*(2 + (-1 + sqrt(2))^(2*floor((1/2)*(-1 + n)))*(-4 + 3*sqrt(2))) + (1 - sqrt(2))^n*(2 - (-1 - sqrt(2))^(2*floor((1/2)*(-1 + n)))*(4 + 3*sqrt(2)))).
From Colin Barker, May 01 2012: (Start)
a(n) = 1 - (-1)^n + (1-sqrt(2))^n + (1+sqrt(2))^n.
G.f.: 2*x*(2-x-2*x^2-x^3)/(1-x)/(1+x)/(1-2*x-x^2). (End)
a(n) = A002203(n) + 1 - (-1)^n. - R. J. Mathar, Oct 08 2016

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User: Sarah-Marie Belcastro

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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A162483 a(n) is the number of perfect matchings of an edge-labeled 2 X (2n+1) Mobius grid graph.

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A162484 a(1) = 2, a(2) = 8; a(n) = 2 a(n - 1) + a(n - 2) - 4*(n mod 2).

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A162485 a(1)=4, a(2)=6; for n > 2, a(n) = 2*a(n-1) + a(n-2) - 4*((n-1) mod 2).

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A162485 a(1)=4, a(2)=6; for n > 2, a(n) = 2a(n-1) + a(n-2) - 4((n-1) mod 2).