OFFSET
0,1
COMMENTS
Alternating sum of all points on the fourth row of the Hosoya triangle composed of Fibonacci polynomials, where F_{0}(n) = 1 and F_{1}(n) = n, hence a(n) = F_{5}(n)/F_{1}(n) for n>0 (see Florez et al. reference, page 7, Table 4 and following sum).
Apart from 8, all terms belong to A217554 because a(n) = (n^2+1)^2 + (n+1)^2 + (n-1)^2 = (n^2+2)^2 - 1. - Bruno Berselli, Jun 04 2018
LINKS
Muniru A Asiru, Table of n, a(n) for n = 0..10000
Rigoberto Florez, Robinson A. Higuita, and Antara Mukherjee, Alternating Sums in the Hosoya Polynomial Triangle, Journal of Integer Sequences, Vol. 17 (2014), Article 14.9.5.
Eric Weisstein's World of Mathematics, Fibonacci Polynomial.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
FORMULA
G.f.: (3 - 7*x + 25*x^2 - 5*x^3 + 8*x^4)/(1-x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = A059100(n)^2 - 1.
Sum_{n>=0} 1/a(n) = 1/6 + coth(Pi)*Pi/4 - coth(sqrt(3)*Pi)*Pi/(4*sqrt(3)). - Amiram Eldar, Feb 24 2023
MAPLE
seq((n^2+2)^2-1, n=0..40); # Muniru A Asiru, Jun 03 2018
MATHEMATICA
Table[n^4 + 4 n^2 + 3, {n, 0, 35}]
LinearRecurrence[{5, -10, 10, -5, 1}, {3, 8, 35, 120, 323}, 40] (* Harvey P. Dale, Mar 04 2021 *)
PROG
(Magma) [n^4+4*n^2+3: n in [0..40]];
(GAP) List([0..40], n -> (n^2+2)^2-1); # Muniru A Asiru, Jun 03 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Vincenzo Librandi, May 31 2018
STATUS
approved