

A078469


Number of different compositions of the ladder graph L_n.


6



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

0,2


COMMENTS

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 2d analog of a partition into integers.  Hugo van der Sanden, Mar 23 2009


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200
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), 222230.
Index entries for linear recurrences with constant coefficients, signature (6,1).


FORMULA

a(n) = 6*a(n1)+a(n2). G.f.: 1+2*x/(16*xx^2).
a(n) = ((3+s)^n(3s)^n)/s, where s = sqrt(10) (assumes a(0) = 0).
Asymptotic to (3+sqrt(10))^n/sqrt(10).  Ralf Stephan, Jan 03 2003
If p[i]=fibonacci(3i) and if A is the Hessenberg matrix of order n defined by: A[i,j]=p[ji+1], (i<=j), A[i,j]=1, (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


MATHEMATICA

Join[{1}, LinearRecurrence[{6, 1}, {2, 12}, 30]] (* Harvey P. Dale, Jul 22 2013 *)


PROG

(MAGMA) I:=[1, 2, 12]; [n le 3 select I[n] else 6*Self(n1)+Self(n2): n in [1..30]]; // Vincenzo Librandi, May 17 2013


CROSSREFS

Cf. A108808, A110476.  Brian Kell, Oct 21 2008
Cf. A152113, A152124.
Sequence in context: A037718 A020049 A020004 * A014351 A074616 A006936
Adjacent sequences: A078466 A078467 A078468 * A078470 A078471 A078472


KEYWORD

nonn,easy


AUTHOR

Ralf Stephan, Jan 02 2003


EXTENSIONS

a(0) changed from 0 to 1 by N. J. A. Sloane, Sep 21 2009, at the suggestion of Hugo van der Sanden


STATUS

approved



