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 A003682 Number of (undirected) Hamiltonian paths in the n-ladder graph K_2 X P_n. 7
 1, 4, 8, 14, 22, 32, 44, 58, 74, 92, 112, 134, 158, 184, 212, 242, 274, 308, 344, 382, 422, 464, 508, 554, 602, 652, 704, 758, 814, 872, 932, 994, 1058, 1124, 1192, 1262, 1334, 1408, 1484, 1562, 1642, 1724, 1808, 1894, 1982, 2072, 2164 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Equals row sums of triangle A144336. - Gary W. Adamson, Sep 18 2008 REFERENCES F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154. LINKS Andrew Howroyd, Table of n, a(n) for n = 1..1000 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 Eric Weisstein's World of Mathematics, Hamiltonian Path Eric Weisstein's World of Mathematics, Ladder Graph Index entries for linear recurrences with constant coefficients, signature (3,-3,1). FORMULA For n>1, a(n) = n^2 - n + 2. Equals binomial transform of [1, 3, 1, 1, -1, 1, -1, 1, ...]. - Gary W. Adamson, Apr 23 2008 G.f.: x(1 + x - x^2 + x^3)/(1-x)^3. - R. J. Mathar, Dec 16 2008 a(n) = floor((n^3 + 2*n)/(n+1)). - Gary Detlefs, Feb 20 2010 Except for the first term, a(n) = 2*n + a(n-1), (with a(1)=4). - Vincenzo Librandi, Dec 06 2010 a(0)=1, a(1)=4, a(2)=8, a(3)=14, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Jun 14 2011 MAPLE a:=n->sum(binomial(2, 2*j)+n, j=0..n): seq(a(n), n=0..46); # Zerinvary Lajos, Feb 22 2007 seq(floor((n^3+2*n)/(n+1)), n=1..47); # Gary Detlefs, Feb 20 2010 MATHEMATICA Join[{1}, Table[n^2 - n + 2, {n, 2, 50}]] (* Harvey P. Dale, Jun 14 2011 *) Join[{1}, LinearRecurrence[{3, -3, 1}, {4, 8, 14}, 50]] (* Harvey P. Dale, Jun 14 2011 *) PROG (PARI) a(n)=if(n>1, n^2-n+2, 1) \\ Charles R Greathouse IV, Jan 05 2018 CROSSREFS Row n=2 of A332307. Equals A002061(n) + 1, n > 1. Cf. A144336. - Gary W. Adamson, Sep 18 2008 Sequence in context: A024398 A054347 A194149 * A333700 A011897 A110895 Adjacent sequences:  A003679 A003680 A003681 * A003683 A003684 A003685 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified January 24 09:05 EST 2021. Contains 340398 sequences. (Running on oeis4.)