|
| |
|
|
A078469
|
|
Number of different compositions of the ladder graph L_n.
|
|
5
|
|
|
|
1, 2, 12, 74, 456, 2810, 17316, 106706, 657552, 4052018, 24969660, 153869978, 948189528, 5843007146, 36006232404, 221880401570, 1367288641824, 8425612252514, 51920962156908, 319951385193962, 1971629273320680
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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 2-d analogue of a partition into integers. [Hugo van der Sanden, Mar 23 2009]
|
|
|
REFERENCES
|
J. N. Ridley and M. E. Mays, Compositions of unions of graphs, Fib. Quart. 42 (2004), 222-230.
|
|
|
LINKS
|
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
A. Knopfmacher and M. E. Mays, Graph compositions,Integers 1(2001), #A04.
Index to sequences with linear recurrences with constant coefficients, signature (6,1).
|
|
|
FORMULA
|
a(n) = 6*a(n-1)+a(n-2). G.f.: 2*x/(1-6*x-x^2) (assumes a(0) = 0).
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
If p[i]=fibonacci(3i) and if A is the Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n)= det A. [From 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(n-1)+Self(n-2): n in [1..30]]; // Vincenzo Librandi, May 17 2013
|
|
|
CROSSREFS
|
Cf. A108808, A110476. [From 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
|
| |
|
|
