A028470 Number of perfect matchings in graph P_{8} X P_{n}.

1, 1, 34, 153, 2245, 14824, 167089, 1292697, 12988816, 108435745, 1031151241, 8940739824, 82741005829, 731164253833, 6675498237130, 59554200469113, 540061286536921, 4841110033666048, 43752732573098281, 393139145126822985, 3547073578562247994, 31910388243436817641

Offset

0,3

References

F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1048

F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Preliminary version of paper that appeared in Ars Combin. 49 (1998), 129-154.

F. Faase, Counting Hamiltonian cycles in product graphs

F. Faase, Results from the counting program

David Klarner, Jordan Pollack, Domino tilings of rectangles with fixed width, Disc. Math. 32 (1980) 45-52

Per Hakan Lundow, Computation of matching polynomials and the number of 1-factors in polygraphs, Research report, No 12, 1996, Department of Math., Umea University, Sweden.

Per Hakan Lundow, Enumeration of matchings in polygraphs, 1998.

R. J. Mathar, Paving rectangular regions with rectangular tiles,...., arXiv:1311.6135 [math.CO], Table 7.

James A. Sellers, Domino Tilings and Products of Fibonacci and Pell Numbers, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.2

Index entries for linear recurrences with constant coefficients, signature (1, 76, 69, -921, -584, 4019, 829, -7012, 829, 4019, -584, -921, 69, 76, 1, -1).

Formula

Recurrence from Faase web site:

a(1) = 1,

a(2) = 34,

a(3) = 153,

a(4) = 2245,

a(5) = 14824,

a(6) = 167089,

a(7) = 1292697,

a(8) = 12988816,

a(9) = 108435745,

a(10) = 1031151241,

a(11) = 8940739824,

a(12) = 82741005829,

a(13) = 731164253833,

a(14) = 6675498237130,

a(15) = 59554200469113,

a(16) = 540061286536921,

a(17) = 4841110033666048,

a(18) = 43752732573098281,

a(19) = 393139145126822985,

a(20) = 3547073578562247994,

a(21) = 31910388243436817641,

a(22) = 287665106926232833093,

a(23) = 2589464895903294456096,

a(24) = 23333526083922816720025,

a(25) = 210103825878043857266833,

a(26) = 1892830605678515060701072,

a(27) = 17046328120997609883612969,

a(28) = 153554399246902845860302369,

a(29) = 1382974514097522648618420280,

a(30) = 12457255314954679645007780869,

a(31) = 112199448394764215277422176953,

a(32) = 1010618564986361239515088848178, and

a(n) = 153a(n-2) - 7480a(n-4) + 151623a(n-6) - 1552087a(n-8) + 8933976a(n-10) - 30536233a(n-12) + 63544113a(n-14) - 81114784a(n-16) + 63544113a(n-18) - 30536233a(n-20) + 8933976a(n-22) - 1552087a(n-24) + 151623a(n-26) - 7480a(n-28) + 153a(n-30) - a(n-32).

G.f.: (1 -43*x^2 -26*x^3 +360*x^4 +110*x^5 -1033*x^6 +1033*x^8 -110*x^9 -360*x^10 +26*x^11 +43*x^12 -x^14) /(1 -x -76*x^2 -69*x^3 +921*x^4 +584*x^5 -4019*x^6 -829*x^7 +7012*x^8 -829*x^9 -4019*x^10 +584*x^11 +921*x^12 -69*x^13 -76*x^14 -x^15 +x^16). - Sergey Perepechko, Nov 22 2012

Maple

a:= n-> (Matrix(16, (i, j)-> `if` (i=j-1, 1, `if` (i=16, [-1, 1, 76, 69, -921, -584, 4019, 829, -7012][min(j, 18-j)], 0)))^n. <<seq([1292697, 167089, 14824, 2245, 153, 34, 1, 1, 0][min(k, 18-k)], k=1..16)>>)[10, 1]: seq(a(n), n=0..50);  # Alois P. Heinz, Apr 14 2011

Mathematica

a[n_] := Product[2(2+Cos[(2j Pi)/9] + Cos[(2k Pi)/(n+1)]), {k, 1, n/2}, {j, 1, 4}] // Round; Join[{1}, Array[a, 21]] (* Jean-François Alcover, Aug 11 2018; a(0)=1 prepended by Georg Fischer, Apr 17 2020 *)

Prog

(PARI) {a(n) = sqrtint(polresultant(polchebyshev(8, 2, x/2), polchebyshev(n, 2, I*x/2)))} \\ Seiichi Manyama, Apr 13 2020

Crossrefs

Row 8 of array A099390.

Sequence in context: A159744 A192398 A167241 * A221806 A321533 A345157

Adjacent sequences: A028467 A028468 A028469 * A028471 A028472 A028473

Keyword

nonn

Author

Per H. Lundow

Extensions

Added recurrence from Faase's web page. - N. J. A. Sloane, Feb 03 2009

a(0)=1 prepended by Seiichi Manyama, Apr 13 2020

Status

approved