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A078469
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Number of different compositions of the ladder graph L_n.
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6
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1, 2, 12, 74, 456, 2810, 17316, 106706, 657552, 4052018, 24969660, 153869978, 948189528, 5843007146, 36006232404, 221880401570, 1367288641824, 8425612252514, 51920962156908, 319951385193962, 1971629273320680
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OFFSET
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0,2
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COMMENTS
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This is equally the number of partitions of a 2 x n rectangle into connected pieces consisting of unit squares cut along lattice lines, like a 2-d analog of a partition into integers. - Hugo van der Sanden, Mar 23 2009
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..200
Liam Buttitta, On the Number of Compositions of Km X Pn, Journal of Integer Sequences, Vol. 25 (2022), Article 22.4.1
Tanya Khovanova, Recursive Sequences
A. Knopfmacher and M. E. Mays, Graph Compositions. I: Basic Enumeration, Integers 1(2001), #A04.
J. N. Ridley and M. E. Mays, Compositions of unions of graphs, Fib. Quart., 42 (2004), 222-230.
Index entries for linear recurrences with constant coefficients, signature (6,1).
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FORMULA
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a(n) = 6*a(n-1) + a(n-2).
G.f.: 1 + 2*x/(1 - 6*x - x^2).
a(n) = ((3 + s)^n - (3 - s)^n)/s, where s = sqrt(10) (assumes a(0) = 0).
Asymptotic to (3 + sqrt(10))^n/sqrt(10). - Ralf Stephan, Jan 03 2003
Let p[i] = Fibonacci(3*i) and A be the Hessenberg matrix of order n defined by: A[i,j] = p[j-i+1], if i <= j; A[i,j] = -1, if i = j + 1; and A[i,j] = 0, otherwise. Then, for n >= 1, a(n) = det(A). - Milan Janjic, May 08 2010
a(n) = 2*A005668(n), n > 0. - R. J. Mathar, Nov 29 2015
a(n) >= A116694(2,n). - R. J. Mathar, Nov 29 2015
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MATHEMATICA
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Join[{1}, LinearRecurrence[{6, 1}, {2, 12}, 30]] (* Harvey P. Dale, Jul 22 2013 *)
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PROG
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(MAGMA) I:=[1, 2, 12]; [n le 3 select I[n] else 6*Self(n-1)+Self(n-2): n in [1..30]]; // Vincenzo Librandi, May 17 2013
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CROSSREFS
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Cf. A108808, A110476. - Brian Kell, Oct 21 2008
Cf. A152113, A152124.
Sequence in context: A342054 A020049 A020004 * A014351 A074616 A352373
Adjacent sequences: A078466 A078467 A078468 * A078470 A078471 A078472
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KEYWORD
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nonn,easy
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AUTHOR
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Ralf Stephan, Jan 02 2003
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EXTENSIONS
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a(0) changed from 0 to 1 by N. J. A. Sloane, Sep 21 2009, at the suggestion of Hugo van der Sanden
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STATUS
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approved
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