A078469 Number of different compositions of the ladder graph L_n.

1, 2, 12, 74, 456, 2810, 17316, 106706, 657552, 4052018, 24969660, 153869978, 948189528, 5843007146, 36006232404, 221880401570, 1367288641824, 8425612252514, 51920962156908, 319951385193962, 1971629273320680

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 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

Liam Buttitta, On the Number of Compositions of Km X Pn, Journal of Integer Sequences, Vol. 25 (2022), Article 22.4.1.

Tomislav Došlić and Luka Podrug, Sweet division problems: from chocolate bars to honeycomb strips and back, arXiv:2304.12121 [math.CO], 2023.

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).

Formula

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

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. - Brian Kell, Oct 21 2008

Cf. A152113, A152124.

Sequence in context: A342054 A020049 A020004 * A014351 A360318 A074616

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