

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



FORMULA

a(n) = 6*a(n1) + a(n2).
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[ji+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


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



KEYWORD

nonn,easy


AUTHOR



EXTENSIONS



STATUS

approved



