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

%I

%S 1,1,2,10,5,7,28,12,15,55,22,26,91,35,40,136,51,57,190,70,77,253,92,

%T 100,325,117,126,406,145,155,496,176,187,595,210,222,703,247,260,820,

%U 287,301,946,330,345,1081,376,392,1225,425,442,1378,477,495,1540,532

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

%C Apart from the first term, this is the same as A027626. [_Bruno Berselli_, Feb 24 2014]

%C This is the sequence of the fourth column of the triangle A107711.

%C a(n) is the numerator of (n+1)*(n+2)/6. - _Altug Alkan_, Apr 19 2018

%H Vincenzo Librandi, <a href="/A234041/b234041.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 G.f.: (1+x+2*x^2+7*x^3+2*x^4+x^5+x^6)/(1-x^3)^3.

%F a(n) = A107711(n+3,3) for n >= 0.

%F a(n) = (2+(-1)^(n+floor((n+1)/3)))*(n+1)*(n+2)/6. [_Bruno Berselli_, Feb 24 2014]

%e a(6) = binomial(8,2) = 28 (example for n == 0 (mod 3)),

%e a(7) = binomial(9,2)/3 = 3*4 = 12 (example for n == 1 (mod 3)),

%e a(8) = binomial(10,2)/3 = 5*3 = 15 (example for n == 2 (mod 3)).

%t Table[Binomial[n + 2, 2] GCD[n + 3, 3]/3, {n, 0, 60}] (* _Bruno Berselli_, Feb 24 2014 *)

%t 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 (PARI) a(n) = numerator((n+1)*(n+2)/6); \\ _Altug Alkan_, Apr 19 2018

%Y Cf. A027626, A107711, A026741 (third column of A107711), A109007 (gcd(n,3)).

%K nonn,easy

%O 0,3

%A _Wolfdieter Lang_, Feb 24 2014

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Last modified June 16 20:32 EDT 2021. Contains 345069 sequences. (Running on oeis4.)