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A234041 a(n) = binomial(n+2,2)*gcd(n,3)/3, n >= 0.

%I A234041 #37 Aug 11 2022 08:53:54
%S A234041 1,1,2,10,5,7,28,12,15,55,22,26,91,35,40,136,51,57,190,70,77,253,92,
%T A234041 100,325,117,126,406,145,155,496,176,187,595,210,222,703,247,260,820,
%U A234041 287,301,946,330,345,1081,376,392,1225,425,442,1378,477,495,1540,532
%N A234041 a(n) = binomial(n+2,2)*gcd(n,3)/3, n >= 0.
%C A234041 Apart from the first term, this is the same as A027626. - _Bruno Berselli_, Feb 24 2014
%C A234041 This is the sequence of the fourth column of the triangle A107711.
%H A234041 Vincenzo Librandi, <a href="/A234041/b234041.txt">Table of n, a(n) for n = 0..1000</a>
%H A234041 <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 A234041 G.f.: (1+x+2*x^2+7*x^3+2*x^4+x^5+x^6)/(1-x^3)^3.
%F A234041 a(n) = A107711(n+3,3) for n >= 0.
%F A234041 a(n) = (2+(-1)^(n+floor((n+1)/3)))*(n+1)*(n+2)/6. - _Bruno Berselli_, Feb 24 2014
%F A234041 a(n) is the numerator of (n+1)*(n+2)/6. - _Altug Alkan_, Apr 19 2018
%F A234041 Sum_{n>=0} 1/a(n) = 6 - 4*Pi/(3*sqrt(3)). - _Amiram Eldar_, Aug 11 2022
%e A234041 a(6) = binomial(8,2) = 28 (example for n == 0 (mod 3)),
%e A234041 a(7) = binomial(9,2)/3 = 3*4 = 12 (example for n == 1 (mod 3)),
%e A234041 a(8) = binomial(10,2)/3 = 5*3 = 15 (example for n == 2 (mod 3)).
%t A234041 Table[Binomial[n + 2, 2] GCD[n + 3, 3]/3, {n, 0, 60}] (* _Bruno Berselli_, Feb 24 2014 *)
%t A234041 CoefficientList[Series[(1 + x + 2 x^2 + 7 x^3 + 2 x^4 + x^5 + x^6)/(1 - x^3)^3, {x, 0, 60}], x] (* _Vincenzo Librandi_, Feb 26 2014 *)
%o A234041 (PARI) a(n) = numerator((n+1)*(n+2)/6); \\ _Altug Alkan_, Apr 19 2018
%Y A234041 Cf. A027626, A107711, A026741 (third column of A107711), A109007 (gcd(n,3)).
%K A234041 nonn,easy
%O A234041 0,3
%A A234041 _Wolfdieter Lang_, Feb 24 2014