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
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