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 A108808 Number of compositions of grid graph G_{3,n} = P_3 X P_n. 7
 4, 74, 1434, 27780, 538150, 10424872, 201947094, 3912050356, 75782907270, 1468040672696, 28438383992230, 550898690444420, 10671821831261942, 206730898391393192, 4004720564629102582, 77578083032366404308, 1502816206487087179878, 29112043791259796460440 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 REFERENCES Reddy, V. and Skiena, S. "Frequencies of Large Distances in Integer Lattices." Technical Report, Department of Computer Science. Stony Brook, NY: State University of New York, Stony Brook, 1989. [Background] Skiena, S. "Grid Graphs." Section 4.2.4 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 147-148, 1990. [Background] LINKS Robert Israel, Table of n, a(n) for n = 1..769 J. N. Ridley and M. E. Mays, Compositions of unions of graphs, Fib. Quart. 42 (2004), 222-230. Frank Simon, Algebraic Methods for Computing the Reliability of Networks, Dissertation, Doctor Rerum Naturalium (Dr. rer. nat.), Fakultät Mathematik und Naturwissenschaften der Technischen Universität Dresden, 2012, Table 6.12. - From N. J. A. Sloane, Jan 04 2013 Eric Weisstein's World of Mathematics, Grid Graph. [Background] Index entries for linear recurrences with constant coefficients, signature (23,-75,91,6,-4). FORMULA From Brian Kell, Oct 20 2008: (Start) a(n) = z * M^(n-1) * w, where z is the 1 x 6 row vector [ 1 ... 1 ], M is the 6 x 6 matrix [[ 2, 3, 3, 3, 4, 5 ], [ 3, 4, 5, 5, 6, 6 ], [ 1, 0, 2, 0, 0, 0 ], [ 3, 5, 5, 4, 6, 6 ], [ 2, 1, 4, 1, 2, 0 ], [ 2, 5, 2, 5, 6, 8 ]], and w is the 6 x 1 column vector [[ 1 ], [ 1 ], [ 0 ], [ 1 ], [ 0 ], [ 1 ]] (End) G.f.: 2*x*(x-2)*(x^3-6*x^2+4*x-1) / (4*x^5-6*x^4-91*x^3+75*x^2-23*x+1). - Colin Barker, May 14 2013 MAPLE z:= <1|1|1|1|1|1>: w:= <1, 1, 0, 1, 0, 1>: M:= Matrix([[ 2, 3, 3, 3, 4, 5 ], [ 3, 4, 5, 5, 6, 6 ], [ 1, 0, 2, 0, 0, 0 ], [ 3, 5, 5, 4, 6, 6 ], [ 2, 1, 4, 1, 2, 0 ], [ 2, 5, 2, 5, 6, 8 ]]): seq(z . M^i . w, i=0..31); # Robert Israel, Dec 03 2015 CROSSREFS Cf. A078469, A110476, A221157 (Grid graph G_{4,n}). Sequence in context: A232371 A114878 A131359 * A358568 A139112 A100865 Adjacent sequences: A108805 A108806 A108807 * A108809 A108810 A108811 KEYWORD nonn,easy AUTHOR N. J. A. Sloane, Jul 09 2005 EXTENSIONS a(4) corrected and a(5)-a(7) computed by Brian Kell, May 20 2008 a(8) - a(11) from Brian Kell, Oct 20 2008 a(12)-a(18) added from Frank Simon's thesis by N. J. A. Sloane, Jan 04 2013 STATUS approved

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Last modified September 26 02:15 EDT 2023. Contains 365649 sequences. (Running on oeis4.)