|
|
A003696
|
|
Number of spanning trees in P_4 X P_n.
|
|
8
|
|
|
1, 56, 2415, 100352, 4140081, 170537640, 7022359583, 289143013376, 11905151192865, 490179860527896, 20182531537581071, 830989874753525760, 34214941811800329425, 1408756312731277540744
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
Also number of domino tilings of the 7 X (2n-1) rectangle with upper left corner removed. - Alois P. Heinz, Apr 14 2011
A linear divisibility sequence of order 8; a(n) divides a(m) whenever n divides m. It is the product of a 2nd-order Lucas sequence and a 4th-order linear divisibility sequence. - Peter Bala, Apr 27 2014
|
|
REFERENCES
|
F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.
|
|
LINKS
|
|
|
FORMULA
|
a(1) = 1,
a(2) = 56,
a(3) = 2415,
a(4) = 100352,
a(5) = 4140081,
a(6) = 170537640,
a(7) = 7022359583,
a(8) = 289143013376 and
a(n) = 56a(n-1) - 672a(n-2) + 2632a(n-3) - 4094a(n-4) + 2632a(n-5) - 672a(n-6) + 56a(n-7) - a(n-8).
G.f.: x(x^6-49x^4+112x^3-49x^2+1) / (x^8-56x^7 +672x^6-2632x^5 +4094x^4 -2632x^3 +672x^2-56x+1). - Paul Raff, Mar 06 2009
a(n) = Resultant( U(3,(x-4)/2),U(n-1,x/2) ), where U(n,x) denotes the Chebyshev polynomial of the second kind. The polynomial U(3,(x-4)/2) = x^3 - 12*x^2 + 46*x - 56 (see A159764) has zeros z_1 = 4, z_2 = 4 + sqrt(2) and z_3 = 4 - sqrt(2). Hence a(n) = U(n-1,2)*U(n-1,1/2*(4 + sqrt(2)))*U(n-1,1/2*(4 - sqrt(2))).
|
|
MAPLE
|
seq(resultant(simplify(ChebyshevU(3, (x-4)*(1/2))), simplify(ChebyshevU(n-1, (1/2)*x)), x), n = 1 .. 14); # Peter Bala, Apr 27 2014
|
|
MATHEMATICA
|
LinearRecurrence[{56, -672, 2632, -4094, 2632, -672, 56, -1}, {1, 56, 2415, 100352, 4140081, 170537640, 7022359583, 289143013376}, 20] (* Jean-François Alcover, Feb 28 2020 *)
|
|
PROG
|
(PARI) {a(n) = polresultant((x-4)*(x^2-8*x+14), polchebyshev(n-1, 2, x/2))}; /* Michael Somos, Oct 31 2022 */
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|