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A217367 a(n) = ((n+7) / gcd(n+7,4)) * (n / gcd(n,4)).

%I A217367 #42 Oct 31 2024 06:46:59
%S A217367 0,2,9,15,11,15,39,49,30,36,85,99,57,65,147,165,92,102,225,247,135,
%T A217367 147,319,345,186,200,429,459,245,261,555,589,312,330,697,735,387,407,
%U A217367 855,897,470,492,1029,1075,561,585,1219,1269,660,686,1425,1479,767,795,1647
%N A217367 a(n) = ((n+7) / gcd(n+7,4)) * (n / gcd(n,4)).
%C A217367 The 7th sequence (p=7) of the family A060819(n)*A060819(n+p).
%H A217367 Vincenzo Librandi, <a href="/A217367/b217367.txt">Table of n, a(n) for n = 0..1000</a>
%H A217367 Wikipedia, <a href="https://en.wikipedia.org/wiki/Quasi-polynomial">Quasi-polynomial</a>.
%H A217367 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (3,-6,10,-12,12,-10,6,-3,1).
%F A217367 a(n) = n*(n+7)/(2*mod(1 + floor(n/2), 2) + 2).
%F A217367 G.f.: x*(2 + 3*x + 6*x^4 - 6*x^5 + 4*x^6 - 3*x^7)/(1 - x + x^2 - x^3)^3.
%F A217367 From _Peter Bala_, Aug 07 2022: (Start)
%F A217367 a(n) = numerator of n*(n+7)/4.
%F A217367 a(n) is quasi-polynomial in n: if p(n) = n*(n+7)/4 then a(4*n) = p(4*n), a(4*n+1) = p(4*n+1), a(4*n+2) = 2*p(4*n+2) and a(4*n+3) = 2*p(4*n+3) = A303295(n+1) for n >= 1. (End)
%F A217367 Sum_{n>=1} 1/a(n) = 697/735 + Pi/14. - _Amiram Eldar_, Aug 16 2022
%t A217367 a[n_] := n*(n+7)/(2*Mod[1 + Floor[n/2], 2] + 2); Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Oct 01 2012 *)
%t A217367 CoefficientList[Series[x (2 + 3 x + 6 x^4 - 6 x^5 + 4 x^6 - 3 x^7) / (1 - x + x^2 - x^3)^3, {x, 0, 33}], x] (* _Vincenzo Librandi_, Jul 17 2013 *)
%o A217367 (PARI) my(x='x+O('x^50)); Vec(x*(2+3*x+6*x^4-6*x^5+4*x^6-3*x^7)/(1-x+x^2-x^3)^3) \\ _G. C. Greubel_, Sep 20 2018
%o A217367 (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(2+3*x+6*x^4-6*x^5+4*x^6-3*x^7)/(1-x+x^2-x^3)^3)); // _G. C. Greubel_, Sep 20 2018
%Y A217367 Cf. A060819, A181318.
%K A217367 nonn,easy
%O A217367 0,2
%A A217367 _Jean-François Alcover_, Oct 01 2012