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 A003779 Number of spanning trees in P_5 x P_n. 4
 1, 209, 30305, 4140081, 557568000, 74795194705, 10021992194369, 1342421467113969, 179796299139278305, 24080189412483072000, 3225041354570508955681, 431926215138756947267505, 57847355494807961811035009, 7747424602888405489208931601 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Also number of domino tilings of the 9 X (2n-1) rectangle with upper left corner removed. - Alois P. Heinz, Apr 14 2011 A linear divisibility sequence of order 16; a(n) divides a(m) whenever n divides m. It is the product of two 4th-order linear divisibility sequences A143699 and A241606. - Peter Bala, Apr 26 2014 REFERENCES F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154. LINKS P. Raff, Table of n, a(n) for n = 1..200 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 P. Raff, Spanning Trees in Grid Graphs. P. Raff, Analysis of the Number of Spanning Trees of P_5 x P_n. Contains sequence, recurrence, generating function, and more. FORMULA a(n) = 209 a(n-1) - 11936 a(n-2) + 274208 a(n-3) - 3112032 a(n-4) + 19456019 a(n-5) - 70651107 a(n-6) + 152325888 a(n-7) - 196664896 a(n-8) + 152325888 a(n-9) - 70651107 a(n-10) + 19456019 a(n-11) - 3112032 a(n-12) + 274208 a(n-13) - 11936 a(n-14) + 209 a(n-15) - a(n-16) [Modified by Paul Raff, Oct 30 2009] G.f.: -x(x^14-1440x^12+26752x^11 -185889x^10+574750x^9-708928x^8 +708928x^6-574750x^5+185889x^4 -26752x^3+1440x^2-1) / (x^16-209x^15 +11936x^14 -274208x^13+3112032x^12-19456019x^11 +70651107x^10 -152325888x^9 +196664896x^8 -152325888x^7+70651107x^6 -19456019x^5 +3112032x^4-274208x^3+11936x^2-209x+1). From Peter Bala, Apr 26 2014: (Start) a(n) = Resultant(U(4,(x-4)/2),U(n-1,x/2)), where U(n,x) denotes the Chebyshev polynomial of the second kind. The polynomial U(4,(x-4)/2) = 209 - 232*x + 93*x^2 - 16*x^3 + x^4 (see A159764) has zeros z_1 = (9 + sqrt(5))/2, z_2 = (9 - sqrt(5))/2, z_3 = (7 + sqrt(5))/2 and z_4 = (7 - sqrt(5))/2. Thus a(n) = U(n-1,1/2*z_1)*U(n-1,1/2*z_2)*U(n-1,1/2*z_3)*U(n-1,1/2*z_4). a(n) = A143699(n)*A241606(n). (End) MAPLE seq(resultant(simplify(ChebyshevU(4, (x-4)*(1/2))), simplify(ChebyshevU(n-1, (1/2)*x)), x), n = 1 .. 14); # Peter Bala, Apr 27 2014 MATHEMATICA a[n_] := 256^(n-1)*Product[Sin[(h*Pi)/10]^2 + Sin[(k*Pi)/(2*n)]^2, {h, 1, 4}, {k, 1, n-1}]; Table[a[n]//Round, {n, 1, 14}] (* Jean-François Alcover, Apr 28 2014 *) PROG (PARI) Vec(-x*(x^14-1440*x^12+26752*x^11-185889*x^10+574750*x^9-708928*x^8+708928*x^6-574750*x^5+185889*x^4-26752*x^3+1440*x^2-1)/(x^16-209*x^15+11936*x^14-274208*x^13+3112032*x^12-19456019*x^11+70651107*x^10-152325888*x^9+196664896*x^8-152325888*x^7+70651107*x^6-19456019*x^5+3112032*x^4-274208*x^3+11936*x^2-209*x+1)+O(x^99)) \\ Charles R Greathouse IV, Nov 13 2015 CROSSREFS A row of A116469. Bisection of A189005. Cf. A143699, A159764, A241606. Sequence in context: A317463 A029554 A203458 * A268659 A265682 A071379 Adjacent sequences:  A003776 A003777 A003778 * A003780 A003781 A003782 KEYWORD nonn,easy AUTHOR EXTENSIONS Recurrence from Faase's web page added by N. J. A. Sloane, Feb 03 2009 STATUS approved

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Last modified January 19 20:41 EST 2020. Contains 331066 sequences. (Running on oeis4.)