This site is supported by donations to The OEIS Foundation.

 Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS". Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A003775 Number of perfect matchings (or domino tilings) in P_5 X P_2n. 8
 1, 8, 95, 1183, 14824, 185921, 2332097, 29253160, 366944287, 4602858719, 57737128904, 724240365697, 9084693297025, 113956161827912, 1429438110270431, 17930520634652959, 224916047725262248, 2821291671062267585, 35389589910135145793, 443918325373278904936 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 REFERENCES F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154. R. P. Stanley, Enumerative Combinatorics I, p. 292. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..300 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, Results from the counting program David Klarner, Jordan Pollack, Domino tilings of rectangles with fixed width, Disc. Math. 32 (1980) 45-52 R. J. Mathar, Paving rectangular regions with rectangular tiles,...., arXiv:1311.6135 [math.CO], Table 4. 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 (15,-32,15,-1). FORMULA If b(n) denotes the number of perfect matchings (or domino tilings) in P_5 X P_n we have: b(1) = 0, b(2) = 8, b(3) = 0, b(4) = 95, b(5) = 0, b(6) = 1183, b(7) = 0, b(8) = 14824 and b(n) = 15b(n-2) - 32b(n-4) + 15b(n-6) - b(n-8). G.f.: (1-x)*(1-6*x+x^2)/(1-15*x+32*x^2-15*x^3+x^4). Let M be the 4 X 4 matrix |1 0 2 8 | 0 1 0 2 | 2 1 5 21| 1 1 1 8 |; then a(n) = M^n(4, 4). - Philippe Deléham, Aug 08 2003 Lim_{n -> Inf} a(n)/a(n-1) = (3 + Sqrt(5))*(5 + Sqrt(21))/4 = 12.54375443458... - Philippe Deléham, Jun 13 2005 a(n) = ((35+7*sqrt(5)+5*sqrt(21)+sqrt(105))*((3+sqrt(5))*(5+sqrt(21))/4)^n+(35-7*sqrt(5)+5*sqrt(21)-sqrt(105))*((3-sqrt(5))*(5+sqrt(21))/4)^n+(35+7*sqrt(5)-5*sqrt(21)-sqrt(105))*((3+sqrt(5))*(5-sqrt(21))/4)^n+(35-7*sqrt(5)-5*sqrt(21)+sqrt(105))*((3-sqrt(5))*(5-sqrt(21))/4)^n)/140. [Tim Monahan, Aug 13 2011] MAPLE a:= n-> (<<15|-32|15|-1>, <1|0|0|0>, <0|1|0|0>, <0|0|1|0>>^n. <<8, 1, 1, 8>>)[2, 1]: seq(a(n), n=0..20);  # Alois P. Heinz, Sep 24 2011 MATHEMATICA a = 3; b = 5; c = 7; d = a*c; e = b*c; g = a*b*c; f[n_] := Simplify[((e + c*Sqrt[b] + b*Sqrt[d] + Sqrt[g])*((a + Sqrt[b])*(b + Sqrt[d])/4)^n + (e - c*Sqrt[b] + b*Sqrt[d] - Sqrt[g])*((a - Sqrt[b])*(b + Sqrt[d])/4)^n + (e + c*Sqrt[b] - b*Sqrt[d] - Sqrt[g])*((a + Sqrt[b])*(b - Sqrt[d])/4)^n + (e - c*Sqrt[b] - 5*Sqrt[d] + Sqrt[g])*((a - Sqrt[b])*(b - Sqrt[d])/4)^n)/ 140]; Array[f, 17, 0] (* Robert G. Wilson v, Aug 13 2011 *) a[n_] := (MatrixPower[{{15, -32, 15, -1}, {1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}}, n].{8, 1, 1, 8})[[2]]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jan 31 2016, after Alois P. Heinz *) CROSSREFS Row 5 of array A099390. Bisection of A189003. Sequence in context: A080208 A099298 A182648 * A262737 A121785 A229294 Adjacent sequences:  A003772 A003773 A003774 * A003776 A003777 A003778 KEYWORD nonn AUTHOR EXTENSIONS Added recurrence from Faase's web page. - N. J. A. Sloane, Feb 03 2009 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.