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A014523
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Number of Hamiltonian paths in a 4 X (2n+1) grid starting at the lower left corner and finishing in the upper right corner.
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6
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1, 4, 20, 111, 624, 3505, 19676, 110444, 619935, 3479776, 19532449, 109638260, 615414276, 3454402959, 19390027600, 108838828241, 610926955724, 3429215026140, 19248644351551, 108045225087424, 606472354675265, 3404210752374756, 19108292005806324
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OFFSET
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0,2
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LINKS
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Table of n, a(n) for n=0..22.
Belgacem Bouras, A New Characterization of Catalan Numbers Related to Hankel Transforms and Fibonacci Numbers, Journal of Integer Sequences, 16 (2013), #13.3.3.
Karen L. Collins, Lucia B. Krompart, The number of Hamiltonian paths in a rectangular grid, Discrete Mathematics, Volume 169, Issues 1-3, 15 May 1997, Pages 29-38.
Michael Dougherty, Christopher French, Benjamin Saderholm, and Wenyang Qian, Hankel Transforms of Linear Combinations of Catalan Numbers, Journal of Integer Sequences, Vol. 14 (2011), Article 11.5.1.
Index entries for two-way infinite sequences
Index entries for sequences related to graphs, Hamiltonian
Index entries for linear recurrences with constant coefficients, signature (7,-9,7,-1).
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FORMULA
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G.f.: (1-3*x+x^2)/(1-7*x+9*x^2-7*x^3+x^4).
a(n) = 7*a(n-1) - 9*a(n-2) + 7*a(n-3) - a(n-4) = -a(-2-n).
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MATHEMATICA
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CoefficientList[Series[(1 - 3 x + x^2)/(1 - 7 x + 9 x^2 - 7 x^3 + x^4), {x, 0, 50}], x] (* Vincenzo Librandi, Dec 21 2015 *)
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PROG
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(PARI) {a(n)= if(n<-1, -a(-2-n), polcoeff( (1-3*x+x^2)/ (1-7*x+9*x^2-7*x^3+x^4) +x*O(x^n), n))} /* Michael Somos, Jun 14 2003 */
(MAGMA) I:=[1, 4, 20, 111]; [n le 4 select I[n] else 7*Self(n-1)- 9*Self(n-2)+7*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Dec 21 2015
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CROSSREFS
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Cf. A014584.
Sequence in context: A263854 A026156 A025183 * A153299 A239643 A081335
Adjacent sequences: A014520 A014521 A014522 * A014524 A014525 A014526
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane, Dec 11 1999
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EXTENSIONS
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Sequence name clarified by Andrew Howroyd, Dec 20 2015
a(21)-a(22) from Vincenzo Librandi, Dec 21 2015
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STATUS
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approved
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