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%I #41 Oct 16 2023 11:27:05
%S 1,2,10,5,7,28,12,15,55,22,26,91,35,40,136,51,57,190,70,77,253,92,100,
%T 325,117,126,406,145,155,496,176,187,595,210,222,703,247,260,820,287,
%U 301,946,330,345,1081,376,392,1225,425,442,1378,477,495,1540,532,551
%N Denominator of n*(n+5)/((n+2)*(n+3)).
%H Vincenzo Librandi, <a href="/A027626/b027626.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,3,0,0,-3,0,0,1).
%F a(n) = GCD of n-th and (n+1)st tetrahedral numbers (A000292). - _Ross La Haye_, Sep 13 2003
%F G.f.: (1 +2*x +10*x^2 +2*x^3 +x^4 -2*x^5 +x^8)/(1-x^3)^3.
%F a(n) = A234041(n+1) = A107711(n+4,3) = C(n+3,2)*gcd(n+4,3)/3 for n >= 0. See the o.g.f. of A234041. - _Wolfdieter Lang_, Feb 26 2014
%F a(n) = numerator of (n+2)*(n+3)/6. - _Altug Alkan_, Apr 18 2018
%F Sum_{n>=0} 1/a(n) = 5 - 4*Pi/(3*sqrt(3)). - _Amiram Eldar_, Aug 11 2022
%F a(n) = (n + 2)*(n + 3)*(5 - 2*A061347(n+1))/18. - _Stefano Spezia_, Oct 16 2023
%t CoefficientList[Series[(1+2*x+10*x^2+2*x^3+x^4-2*x^5+x^8)/(1-x^3)^3, {x,0,50}], x] (* _Vincenzo Librandi_, Mar 04 2014 *)
%o (Magma) [Denominator(n*(n+5)/((n+2)*(n+3))): n in [0..60]]; // _Vincenzo Librandi_, Mar 04 2014
%o (PARI) a(n) = numerator((n+2)*(n+3)/6); \\ _Altug Alkan_, Apr 18 2018
%o (SageMath) [numerator(binomial(n+3,2)/3) for n in (0..60)] # _G. C. Greubel_, Aug 04, 2022
%Y Cf. A000292, A027625, A061347, A107711, A234041.
%K nonn,frac,easy
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _Vincenzo Librandi_, Mar 04 2014
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