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 A304726 a(n) = n^4 + 4*n^2 + 3. 1
 3, 8, 35, 120, 323, 728, 1443, 2600, 4355, 6888, 10403, 15128, 21315, 29240, 39203, 51528, 66563, 84680, 106275, 131768, 161603, 196248, 236195, 281960, 334083, 393128, 459683, 534360, 617795, 710648, 813603, 927368, 1052675, 1190280, 1340963, 1505528, 1684803 (list; graph; refs; listen; history; text; internal format)
 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 Cf. A058071, A059100, A217554. Subsequence of A005563. Sequence in context: A077291 A192212 A148918 * A226679 A216541 A347896 Adjacent sequences: A304723 A304724 A304725 * A304727 A304728 A304729 KEYWORD nonn,easy AUTHOR Vincenzo Librandi, May 31 2018 STATUS approved

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Last modified February 28 03:01 EST 2024. Contains 370379 sequences. (Running on oeis4.)