OFFSET
0,2
COMMENTS
Kekulé numbers for certain benzenoids.
LINKS
Chai Wah Wu, Table of n, a(n) for n = 0..1000
Paolo Aluffi, Degrees of projections of rank loci, arXiv:1408.1702 [math.AG], 2014. ["After compiling the results of many explicit computations, we noticed that many of the numbers d_{n,r,S} appear in the existing literature in contexts far removed from the enumerative geometry of rank conditions; we owe this surprising (to us) observation to perusal of [Slo14]."]
S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 167).
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).
FORMULA
a(0)=1, a(1)=16, a(2)=110, a(3)=490, a(4)=1666, a(5)=4704, a(6)=11592, a(7)=25740, a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8). - Harvey P. Dale, Aug 07 2013
G.f.: (1 + 8*x + 10*x^2 + 2*x^3) / (1 - x)^8. - Colin Barker, Apr 22 2020
From Amiram Eldar, May 31 2022: (Start)
Sum_{n>=0} 1/a(n) = 3645*sqrt(3)*Pi/28 + 75*Pi^2 - 32805*log(3)/28 - 2245/14.
Sum_{n>=0} (-1)^n/a(n) = 3645*sqrt(3)*Pi/14 - 45*Pi^2/2 - 7680*log(2)/7 - 6065/14. (End)
MAPLE
a:=n->(1/720)*(n+1)*(n+2)^2*(n+3)^2*(n+4)*(3*n+5): seq(a(n), n=0..33);
MATHEMATICA
Table[((n+1)(n+2)^2 (n+3)^2 (n+4)(3n+5))/720, {n, 0, 30}] (* or *) LinearRecurrence[ {8, -28, 56, -70, 56, -28, 8, -1}, {1, 16, 110, 490, 1666, 4704, 11592, 25740}, 30] (* Harvey P. Dale, Aug 07 2013 *)
PROG
(Python)
from itertools import islice
def A107908_generator():
m = [21, -13, 3]+[1]*5
yield m[-1]
while True:
for i in range(7):
m[i+1]+= m[i]
yield m[-1]
list(islice(A107908_generator(), 0, 50, 1)) # Chai Wah Wu, Nov 14 2014
(PARI) Vec((1 + 8*x + 10*x^2 + 2*x^3) / (1 - x)^8 + O(x^30)) \\ Colin Barker, Apr 22 2020
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, Jun 12 2005
STATUS
approved