Search: id:A004253
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%I A004253 M3553
%S A004253 1,4,19,91,436,2089,10009,47956,229771,1100899,5274724,25272721,
%T A004253 121088881,580171684,2779769539,13318676011,63813610516,305749376569,
%U A004253 1464933272329,7018916985076,33629651653051,161129341280179
%N A004253 a(n) = 5a(n-1) - a(n-2).
%C A004253 Number of domino tilings in K_3 X P_2n (or in S_4 X P_2n).
%C A004253 Number of perfect matchings in graph C_{3} X P_{2n}.
%C A004253 Number of perfect matchings in S_4 X P_2n.
%C A004253 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
%C A004253 a(n) = L(n,5), where L is defined as in A108299; see also A030221 for 
               L(n,-5). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Jun 01 2005
%C A004253 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
%C A004253 (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
%D A004253 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A004253 F. Faase, On the number of specific spanning subgraphs of the graphs 
               G X P_n, Ars Combin. 49 (1998), 129-154.
%D A004253 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.
%D A004253 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.
%D A004253 F. M. van Lamoen, Square wreaths around hexagons, Forum Geometricorum, 
               6 (2006) 311-325.
%H A004253 T. D. Noe, <a href="b004253.txt">Table of n, a(n) for n=1..200</a>
%H A004253 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A004253 F. Faase, <a href="http://www.iwriteiam.nl/Cpaper.zip">On the number 
               of specific spanning subgraphs of the graphs G X P_n</a>, Preliminary 
               version of paper that appeared in Ars Combin. 49 (1998), 129-154.
%H A004253 F. Faase, <a href="http://www.iwriteiam.nl/counting.html">Counting Hamilton 
               cycles in product graphs</a>
%H A004253 F. Faase, <a href="http://www.iwriteiam.nl/Cresults.html">Results from 
               the counting program</a>
%H A004253 F. Faase, <a href="http://home.wxs.nl/~faase009/counting.html">Counting 
               Hamilton cycles in product graphs</a>
%H A004253 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=422">
               Encyclopedia of Combinatorial Structures 422</a>
%H A004253 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A004253 Per Hakan Lundow, <a href="http://www.theophys.kth.se/~phl/Text/1factors2.ps.gz">
               Enumeration of matchings in polygraphs</a>, 1998.
%H A004253 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A004253 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A004253 James A. Sellers, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Domino Tilings and Products of Fibonacci and Pell Numbers</a>, Journal 
               of Integer Sequences, Vol. 5 (2002), Article 02.1.2
%H A004253 F. M. van Lamoen, <a href="http://forumgeom.fau.edu/FG2006volume6/FG200637index.html">
               Article in Forum Geometricorum</a>
%H A004253 <a href="Sindx_Do.html#domino">Index entries for sequences related to 
               dominoes</a>
%F A004253 G.f.: (1 - x) / (1 - 5x + x^2 ).
%F A004253 For n>1 a(n)=b(n)+b(n-1) with b(n) as in A005386. - Floor van Lamoen, 
               Dec 13 2006
%F A004253 a(n) ~ (1/2 + 1/14*sqrt(21))*(1/2*(5 + sqrt(21)))^n - Joe Keane (jgk(AT)jgk.org), 
               May 16 2002
%F A004253 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
%F A004253 For n>0, a(n)a(n+3) = 15 + a(n+1)a(n+2). - R. Stephan, May 29 2004
%F A004253 a(n)=sum{k=0..n, binomial(n+k, 2k)3^k} - Paul Barry (pbarry(AT)wit.ie), 
               Jul 26 2004
%F A004253 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
%F A004253 [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
%p A004253 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
%p A004253 A004253:=-(-1+z)/(1-5*z+z**2); [S. Plouffe in his 1992 dissertation.]
%o A004253 (Other) sage: [lucas_number1(n,5,1)-lucas_number1(n-1,5,1) for n in xrange(1, 
               23)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 10 
               2009]
%Y A004253 Cf. A030221, A003501.
%Y A004253 Partial sums are in A004254.
%Y A004253 Row 5 of array A094954.
%Y A004253 Cf. A004254.
%Y A004253 Sequence in context: A015530 A010907 A087449 this_sequence A151253 A121179 
               A131552
%Y A004253 Adjacent sequences: A004250 A004251 A004252 this_sequence A004254 A004255 
               A004256
%K A004253 nonn
%O A004253 1,2
%A A004253 Frans Faase (Frans_LiXia(AT)wxs.nl), Per Hakan Lundow (phl(AT)theophys.kth.se)
%E A004253 Additional comments from James Sellers and N. J. A. Sloane (njas(AT)research.att.com), 
               May 03, 2002
%E A004253 More terms from Ray Chandler (rayjchandler(AT)sbcglobal.net), Nov 17 
               2003

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