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A304726 a(n) = n^4 + 4*n^2 + 3.

%I A304726 #66 Feb 16 2025 08:33:54
%S A304726 3,8,35,120,323,728,1443,2600,4355,6888,10403,15128,21315,29240,39203,
%T A304726 51528,66563,84680,106275,131768,161603,196248,236195,281960,334083,
%U A304726 393128,459683,534360,617795,710648,813603,927368,1052675,1190280,1340963,1505528,1684803
%N A304726 a(n) = n^4 + 4*n^2 + 3.
%C A304726 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).
%C A304726 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
%H A304726 Muniru A Asiru, <a href="/A304726/b304726.txt">Table of n, a(n) for n = 0..10000</a>
%H A304726 Rigoberto Florez, Robinson A. Higuita, and Antara Mukherjee, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Mukherjee/mukh2.html">Alternating Sums in the Hosoya Polynomial Triangle</a>, Journal of Integer Sequences, Vol. 17 (2014), Article 14.9.5.
%H A304726 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FibonacciPolynomial.html">Fibonacci Polynomial</a>.
%H A304726 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A304726 G.f.: (3 - 7*x + 25*x^2 - 5*x^3 + 8*x^4)/(1-x)^5.
%F A304726 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
%F A304726 a(n) = A059100(n)^2 - 1.
%F A304726 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
%p A304726 seq((n^2+2)^2-1,n=0..40); # _Muniru A Asiru_, Jun 03 2018
%t A304726 Table[n^4 + 4 n^2 + 3, {n, 0, 35}]
%t A304726 LinearRecurrence[{5,-10,10,-5,1},{3,8,35,120,323},40] (* _Harvey P. Dale_, Mar 04 2021 *)
%o A304726 (Magma) [n^4+4*n^2+3: n in [0..40]];
%o A304726 (GAP) List([0..40], n -> (n^2+2)^2-1); # _Muniru A Asiru_, Jun 03 2018
%Y A304726 Cf. A058071, A059100, A217554.
%Y A304726 Subsequence of A005563.
%K A304726 nonn,easy
%O A304726 0,1
%A A304726 _Vincenzo Librandi_, May 31 2018