1,2

Number of domino tilings in K_3 X P_2n (or in S_4 X P_2n).

Number of perfect matchings in graph C_{3} X P_{2n}.

Number of perfect matchings in S_4 X P_2n.

In general, sum{k=0..n, binomial(2n-k,k)j^(n-k)}=(-1)^n*U(2n,I*sqrt(j)/2), I=sqrt(-1). - Paul Barry (pbarry(AT)wit.ie), Mar 13 2005

a(n) = L(n,5), where L is defined as in A108299; see also A030221 for L(n,-5). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jun 01 2005

Number of 01-avoiding words of length n on alphabet {0,1,2,3,4} which do not end in 0. (e.g. n=2, we have 02, 03, 04, 11, 12, 13, 14, 21, 22, 23, 24, 31, 32, 33, 34, 41, 42, 43, 44) - Tanya Khovanova (tanyakh(AT)yahoo.com), Jan 10 2007

(Sqrt(21)+5))/2 = 4.7912878... = exp ArcCosh(5/2) = 4 + 3/4 + 3/(4*19) + 3/(19*91) + 3/(91*436)... - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 18 2007

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures}, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

F. A. Haight, On a generalization of Pythagoras' theorem, pp. 73-77 of J. C. Butcher, editor, A Spectrum of Mathematics. Auckland University Press, 1971.

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

F. M. van Lamoen, Square wreaths around hexagons, Forum Geometricorum, 6 (2006) 311-325.

T. D. Noe, Table of n, a(n) for n=1..200

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures}, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

Tanya Khovanova, Recursive Sequences

F. Faase, Counting Hamilton cycles in product graphs

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 422

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

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

F. M. van Lamoen, Article in Forum Geometricorum

Index entries for sequences related to dominoes

G.f.: (1 - x) / (1 - 5x + x^2 ).

For n>1 a(n)=b(n)+b(n-1) with b(n) as in A005386. - Floor van Lamoen, Dec 13 2006

a(n) ~ (1/2 + 1/14*sqrt(21))*(1/2*(5 + sqrt(21)))^n - Joe Keane (jgk(AT)jgk.org), May 16 2002

Let q(n, x)=sum(i=0, n, x^(n-i)*binomial(2*n-i, i)); then q(n, 3)=a(n) - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 10 2002

For n>0, a(n)a(n+3) = 15 + a(n+1)a(n+2). - R. Stephan, May 29 2004

a(n)=sum{k=0..n, binomial(n+k, 2k)3^k} - Paul Barry (pbarry(AT)wit.ie), Jul 26 2004

a(n)=(-1)^n*U(2n, I*sqrt(3)/2), U(n, x) Chebyshev polynomial of second kind, I=sqrt(-1); - Paul Barry (pbarry(AT)wit.ie), Mar 13 2005

[a(n), A004254(n)] = the 2 X 2 matrix [1,3; 1,4]^n * [1,0]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 19 2008

a[0]:=1: a[1]:=1: for n from 2 to 26 do a[n]:=5*a[n-1]-a[n-2] od: seq(a[n], n=1..22); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 26 2006

A004253:=-(-1+z)/(1-5*z+z**2); [Conjectured by S. Plouffe in his 1992 dissertation.]

Cf. A030221, A003501.

Partial sums are in A004254.

Row 5 of array A094954.

Cf. A004254.

Adjacent sequences: A004250 A004251 A004252 this_sequence A004254 A004255 A004256

Sequence in context: A015530 A010907 A087449 this_sequence A121179 A131552 A122369

nonn

Frans Faase (Frans_LiXia(AT)wxs.nl), Per Hakan Lundow (phl(AT)theophys.kth.se)

Additional comments from James Sellers and njas, May 03, 2002

More terms from Ray Chandler (rayjchandler(AT)sbcglobal.net), Nov 17 2003