cp's OEIS Frontend

A078469 Number of different compositions of the ladder graph L_n.

%I A078469 #54 Jan 05 2025 19:51:37
%S A078469 1,2,12,74,456,2810,17316,106706,657552,4052018,24969660,153869978,
%T A078469 948189528,5843007146,36006232404,221880401570,1367288641824,
%U A078469 8425612252514,51920962156908,319951385193962,1971629273320680
%N A078469 Number of different compositions of the ladder graph L_n.
%C A078469 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
%H A078469 Vincenzo Librandi, <a href="/A078469/b078469.txt">Table of n, a(n) for n = 0..200</a>
%H A078469 Liam Buttitta, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL25/Buttitta/but3.html">On the Number of Compositions of Km X Pn</a>, Journal of Integer Sequences, Vol. 25 (2022), Article 22.4.1.
%H A078469 Tomislav Došlić and Luka Podrug, <a href="https://arxiv.org/abs/2304.12121">Sweet division problems: from chocolate bars to honeycomb strips and back</a>, arXiv:2304.12121 [math.CO], 2023.
%H A078469 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A078469 A. Knopfmacher and M. E. Mays, <a href="http://www.emis.de/journals/INTEGERS/papers/b4/b4.Abstract.html">Graph Compositions. I: Basic Enumeration</a>, Integers 1(2001), #A04.
%H A078469 J. N. Ridley and M. E. Mays, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/42-3/Ridley-Mays-scanned.pdf">Compositions of unions of graphs</a>, Fib. Quart., 42 (2004), 222-230.
%H A078469 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,1).
%F A078469 a(n) = 6*a(n-1) + a(n-2).
%F A078469 G.f.: 1 + 2*x/(1 - 6*x - x^2).
%F A078469 a(n) = ((3 + s)^n - (3 - s)^n)/s, where s = sqrt(10) (assumes a(0) = 0).
%F A078469 Asymptotic to (3 + sqrt(10))^n/sqrt(10). - _Ralf Stephan_, Jan 03 2003
%F A078469 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
%F A078469 a(n) = 2*A005668(n), n > 0. - _R. J. Mathar_, Nov 29 2015
%F A078469 a(n) >= A116694(2,n). - _R. J. Mathar_, Nov 29 2015
%t A078469 Join[{1},LinearRecurrence[{6,1},{2,12},30]] (* _Harvey P. Dale_, Jul 22 2013 *)
%o A078469 (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
%Y A078469 Cf. A108808, A110476. - _Brian Kell_, Oct 21 2008
%Y A078469 Cf. A152113, A152124.
%K A078469 nonn,easy
%O A078469 0,2
%A A078469 _Ralf Stephan_, Jan 02 2003
%E A078469 a(0) changed from 0 to 1 by _N. J. A. Sloane_, Sep 21 2009, at the suggestion of _Hugo van der Sanden_