A108808 Number of compositions of grid graph G_{3,n} = P_3 X P_n.

4, 74, 1434, 27780, 538150, 10424872, 201947094, 3912050356, 75782907270, 1468040672696, 28438383992230, 550898690444420, 10671821831261942, 206730898391393192, 4004720564629102582, 77578083032366404308, 1502816206487087179878, 29112043791259796460440

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