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 A275496 #94 Aug 21 2022 04:18:47
%S A275496 0,1,36,153,528,1225,2628,4753,8256,13041,20100,29161,41616,56953,
%T A275496 77028,101025,131328,166753,210276,260281,320400,388521,468996,559153,
%U A275496 664128,780625,914628,1062153,1230096,1413721,1620900,1846081,2098176,2370753,2673828
%N A275496 a(n) = n^2*(2*n^2 + (-1)^n).
%C A275496 All terms of this sequence are triangular numbers. Graphically, for each term of the sequence, one corner of the square of squares (4th power) will be part of the corresponding triangle's hypotenuse if the term is an odd number. Otherwise, it will not be part of it.
%C A275496 a(A000129(n)) is a square triangular number.
%C A275496 a(2^((A000043(n) - 1)/2)) - 2^A000043(n) is a perfect number.
%H A275496 Colin Barker and Daniel Poveda Parrilla, <a href="/A275496/b275496.txt">Table of n, a(n) for n = 0..46340</a> [n = 1 through 1000 by Colin Barker, Aug 02 2016; and n=1001 to 46340 by Daniel Poveda Parrilla, Aug 04 2016]
%H A275496 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-6,0,6,-2,-2,1).
%F A275496 a(n) = n^4 + Sum_{k=0..(n^2 - (n mod 2))} 2k.
%F A275496 a(n) = A275543(n)*(n^2).
%F A275496 From _Colin Barker_, Aug 01 2016 and Aug 04 2016: (Start)
%F A275496 a(n) = n^2*(2*n^2 + (-1)^n).
%F A275496 a(n) = 2*n^4 + n^2 for n even.
%F A275496 a(n) = 2*n^4 - n^2 for n odd.
%F A275496 G.f.: x*(1 +34*x +79*x^2 +156*x^3 +79*x^4 +34*x^5 +x^6) / ((1-x)^5*(1+x)^3).
%F A275496 (End)
%F A275496 a(n) = n^2*A000217(2n-1) + 2n*A000217(n-(n mod 2)) for n > 0.
%F A275496 E.g.f.: x*(2*(1 + 7*x + 6*x^2 + x^3)*exp(x) - exp(-x)). - _G. C. Greubel_, Aug 05 2016
%F A275496 a(n) = A000217(A077221(n)).
%F A275496 a(n) = (A001844(A077221(n)) - 1)/4.
%F A275496 Sum_{n>=1} 1/a(n) = 1 - Pi^2/12 + (tan(c) - coth(c))*c, where c = Pi/(2*sqrt(2)) is A093954. - _Amiram Eldar_, Aug 21 2022
%e A275496 a(5) = 5^4 + Sum_{k=0..(5^2 - (5 mod 2))} 2k = 625 + Sum_{k=0..(25 - 1)} 2k = 625 + 600 = 1225.
%e A275496 a(12) = 12^4 + Sum_{k=0..(12^2 - (12 mod 2))} 2k = 20736 + Sum_{k=0..(144 - 0)} 2k = 20736 + 20880 = 41616.
%t A275496 Table[n^2 ((-1)^n + 2 n^2), {n, 0, 34}] (* or *)
%t A275496 CoefficientList[Series[x (1 + 34 x + 79 x^2 + 156 x^3 + 79 x^4 + 34 x^5 +
%t A275496 x^6)/((1 - x)^5 (1 + x)^3), {x, 0, 34}], x] (* _Michael De Vlieger_, Aug 01 2016 *)
%t A275496 LinearRecurrence[{2,2,-6,0,6,-2,-2,1},{0,1,36,153,528,1225,2628,4753},40] (* _Harvey P. Dale_, Sep 10 2016 *)
%o A275496 (PARI) a(n)=n=n^2; if(n%2,2*n-1,2*n+1)*n \\ _Charles R Greathouse IV_, Jul 30 2016
%o A275496 (PARI) concat(0, Vec(x*(1+34*x+79*x^2+156*x^3+79*x^4+34*x^5+x^6)/((1-x)^5*(1+x)^3) + O(x^100))) \\ _Colin Barker_, Aug 01 2016
%Y A275496 Cf. A000040, A000129, A000217, A001110, A077221, A093954, A275543.
%K A275496 nonn,easy
%O A275496 0,3
%A A275496 _Daniel Poveda Parrilla_, Jul 30 2016
%E A275496 New name from _Colin Barker_, Aug 04 2016