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a(n) = n^2*(2*n^2 + (-1)^n).
4

%I #94 Aug 21 2022 04:18:47

%S 0,1,36,153,528,1225,2628,4753,8256,13041,20100,29161,41616,56953,

%T 77028,101025,131328,166753,210276,260281,320400,388521,468996,559153,

%U 664128,780625,914628,1062153,1230096,1413721,1620900,1846081,2098176,2370753,2673828

%N a(n) = n^2*(2*n^2 + (-1)^n).

%C 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 a(A000129(n)) is a square triangular number.

%C a(2^((A000043(n) - 1)/2)) - 2^A000043(n) is a perfect number.

%H 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 <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 a(n) = n^4 + Sum_{k=0..(n^2 - (n mod 2))} 2k.

%F a(n) = A275543(n)*(n^2).

%F From _Colin Barker_, Aug 01 2016 and Aug 04 2016: (Start)

%F a(n) = n^2*(2*n^2 + (-1)^n).

%F a(n) = 2*n^4 + n^2 for n even.

%F a(n) = 2*n^4 - n^2 for n odd.

%F 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 (End)

%F a(n) = n^2*A000217(2n-1) + 2n*A000217(n-(n mod 2)) for n > 0.

%F E.g.f.: x*(2*(1 + 7*x + 6*x^2 + x^3)*exp(x) - exp(-x)). - _G. C. Greubel_, Aug 05 2016

%F a(n) = A000217(A077221(n)).

%F a(n) = (A001844(A077221(n)) - 1)/4.

%F 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 a(5) = 5^4 + Sum_{k=0..(5^2 - (5 mod 2))} 2k = 625 + Sum_{k=0..(25 - 1)} 2k = 625 + 600 = 1225.

%e a(12) = 12^4 + Sum_{k=0..(12^2 - (12 mod 2))} 2k = 20736 + Sum_{k=0..(144 - 0)} 2k = 20736 + 20880 = 41616.

%t Table[n^2 ((-1)^n + 2 n^2), {n, 0, 34}] (* or *)

%t CoefficientList[Series[x (1 + 34 x + 79 x^2 + 156 x^3 + 79 x^4 + 34 x^5 +

%t x^6)/((1 - x)^5 (1 + x)^3), {x, 0, 34}], x] (* _Michael De Vlieger_, Aug 01 2016 *)

%t 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 (PARI) a(n)=n=n^2; if(n%2,2*n-1,2*n+1)*n \\ _Charles R Greathouse IV_, Jul 30 2016

%o (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 Cf. A000040, A000129, A000217, A001110, A077221, A093954, A275543.

%K nonn,easy

%O 0,3

%A _Daniel Poveda Parrilla_, Jul 30 2016

%E New name from _Colin Barker_, Aug 04 2016