OFFSET
3,1
COMMENTS
The sequence is not monotone, although it seems to be.
It has two monotone subsequences depending on the parity of n.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 3..100
Artem M. Karavaev, Hamilton Cycles: Flow Problem.
Eric Weisstein's World of Mathematics, Hamiltonian Cycle
Eric Weisstein's World of Mathematics, Torus Grid Graph
FORMULA
a(n) = P(n) + Q(floor(n/2)) if n is even and a(n) = P(n) + R(floor(n/2)) if n is odd, where P(n) = (2*n + 1)*cosh(2*n*arctanh(sqrt(1/3))) - (n/sqrt(3))*sinh(2*n*arctanh(sqrt(1/3))) + cos(Pi*n/2) - sin(Pi*n/2), Q(n) = (4^n - 16*3^n - 4)/3 + 8*2^(n/2)*cos(n*arctan(sqrt(7))) + 4*2^n*cosh(2*n*arctanh(sqrt(2/3))) - 2*cosh(2*n*arctanh(sqrt(1/2))), R(n) = -2*(n + 1)*(2 - (-1)^n).
G.f.: -48*x^2 - 2*x - 17/3 + (1 - x)/(x^2 + 1) + 1/(6*(2*x + 1)) + (x + 1)/(x^2 - 2*x - 1) - 1/((x - 1)^2) + (8 - 4*x^2)/(2*x^4 - x^2 + 1) + (-16 + 62*x)/(x^2 - 4*x + 1)^2 - 2/3/(x + 1) + 1/((x + 1)^2) + (17 + 3*x)/(x^2 - 4*x + 1) + (-2 - 4*x)/(2*x^2 - 4*x - 1) + 2/3/(x - 1) - 1/(6*(2*x - 1)) + (1 - x)/(x^2 + 2*x - 1) + (-2 + 4*x)/(2*x^2 + 4*x - 1) + 16/3/(3*x^2 - 1) + 2*x/(x^2 + 1)^2.
Asympt.: a(n) ~ 2*(2 + sqrt(6))^n if n is even and
a(n) ~ ((1 - 1/(2*sqrt(3)))*n + 1/2)*(2 + sqrt(3))^n if n is odd.
MAPLE
P := n -> (2*n+1)*cosh(2*n*arctanh(sqrt(1/3))) - (n/sqrt(3))*sinh(2*n*arctanh(sqrt(1/3))) + cos(Pi*n/2) - sin(Pi*n/2):
Q := n -> (4^n-16*3^n-4)/3+8*2^(n/2)*cos(n*arctan(sqrt(7))) + 4*2^n*cosh(2*n*arctanh(sqrt(2/3)))-2*cosh(2*n*arctanh(sqrt(1/2))):
R := n -> -2*(n+1)*(2-(-1)^n):
a := n -> expand(P(n)) + (1 - n mod 2)*expand(Q(floor(n/2))) + (n mod 2)*R(floor(n/2)):
seq(a(n), n=3..24);
CROSSREFS
KEYWORD
nonn
AUTHOR
Artem M. Karavaev, Sep 09 2012
STATUS
approved