This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A352115 #35 Mar 11 2022 20:48:34
%S A352115 0,6,16,44,80,146,224,344,480,670,880,1156,1456,1834,2240,2736,3264,
%T A352115 3894,4560,5340,6160,7106,8096,9224,10400,11726,13104,14644,16240,
%U A352115 18010,19840,21856,23936,26214,28560,31116,33744,36594,39520,42680,45920,49406,52976,56804
%N A352115 Partial sums of the even triangular numbers (A014494).
%C A352115 The absolute difference between the n-th partial sum of the odd triangular numbers and the (n-1)-th partial sum of the even triangular numbers is equal to n; see formula.
%C A352115 Partial sums of the even generalized hexagonal numbers. - _Omar E. Pol_, Mar 05 2022
%H A352115 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-4,1,2,-1).
%F A352115 a(n) = Sum_{k=0..n-1} A014494(k) = Sum_{k=0..n-1} (2*k+1)(2*k+1-(-1)^k)/2.
%F A352115 |A352116(n) - a(n-1)| = n.
%F A352115 A352116(n) + a(n-1) = A000447(n), (n >= 1).
%F A352115 From _Stefano Spezia_, Mar 05 2022: (Start)
%F A352115 a(n) = (n + 1)*(4*n^2 + 8*n + 3 - 3*(-1)^n)/6.
%F A352115 G.f.: 2*x*(3 + 2*x + 3*x^2)/((1 - x)^4*(1 + x)^2). (End)
%e A352115 a(0) = 0 because 0 is the first even term in A000217.
%e A352115 a(1) = 6, the sum of the first two even terms in A000217, and so on.
%t A352115 LinearRecurrence[{2, 1, -4, 1, 2, -1}, {0, 6, 16, 44, 80, 146}, 50] (* _Amiram Eldar_, Mar 05 2022 *)
%o A352115 (PARI) te(n) = (2*n+1)*(2*n+1-(-1)^n)/2; \\ A014494
%o A352115 a(n) = sum(k=0, n, te(k)); \\ _Michel Marcus_, Mar 06 2022
%o A352115 (Python)
%o A352115 def A352115(n): return (n + 1)*(2*n*(n+2) + 3*(n%2))//3 # _Chai Wah Wu_, Mar 11 2022
%Y A352115 Cf. A001477, A000217, A000292, A014493, A014494, A352116, A000447.
%K A352115 nonn,easy
%O A352115 0,2
%A A352115 _David James Sycamore_, Mar 05 2022