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A110345 a(n) = n + (n+1) + (n+2) + ... n terms if n is odd, else a(n) = n + (n-1) + (n-2) + ... n terms = n(n+1)/2 = n-th triangular number if n is even.

%I A110345 #15 Sep 11 2022 06:21:51
%S A110345 1,3,12,10,35,21,70,36,117,55,176,78,247,105,330,136,425,171,532,210,
%T A110345 651,253,782,300,925,351,1080,406,1247,465,1426,528,1617,595,1820,666,
%U A110345 2035,741,2262,820,2501,903,2752,990,3015,1081,3290,1176,3577,1275,3876,1378
%N A110345 a(n) = n + (n+1) + (n+2) + ... n terms if n is odd, else a(n) = n + (n-1) + (n-2)  + ... n terms = n(n+1)/2 = n-th triangular number if n is even.
%H A110345 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-3,0,1).
%F A110345 a(2n) = n*(2n+1), a(2n-1) = (2n-1)*(n-1)+(2n-1)^2. - _Stefan Steinerberger_, Jan 24 2006
%F A110345 From _Bruno Berselli_, Mar 19 2012: (Start)
%F A110345 G.f.: x*(1+3x+9x^2+x^3+2x^4)/(1-x^2)^3.
%F A110345 a(n) = n^2-(-1)^n*(n-1)*n/2. (End)
%F A110345 Sum_{n>=1} 1/a(n) = 2 + Pi/(2*sqrt(3)) + log(3*sqrt(3)/16). - _Amiram Eldar_, Sep 11 2022
%e A110345 a(3) = 3+4+5 = 12.
%e A110345 a(6) = 6+5+4+3+2+1 = 21.
%t A110345 For[n = 1, n < 50, n++, If[EvenQ[n], Print[n*(n + 1)/2], Print[n^2 + n*(n - 1)/2]]] (* _Stefan Steinerberger_, Jan 24 2006 *)
%Y A110345 Cf. A000217, A110344.
%K A110345 nonn,easy
%O A110345 1,2
%A A110345 _Amarnath Murthy_, Jul 20 2005
%E A110345 More terms from _Stefan Steinerberger_, Jan 24 2006