A275496 a(n) = n^2*(2*n^2 + (-1)^n).

0, 1, 36, 153, 528, 1225, 2628, 4753, 8256, 13041, 20100, 29161, 41616, 56953, 77028, 101025, 131328, 166753, 210276, 260281, 320400, 388521, 468996, 559153, 664128, 780625, 914628, 1062153, 1230096, 1413721, 1620900, 1846081, 2098176, 2370753, 2673828

Offset

0,3

Comments

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.

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

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

Links

Colin Barker and Daniel Poveda Parrilla, Table of n, a(n) for n = 0..46340 [n = 1 through 1000 by Colin Barker, Aug 02 2016; and n=1001 to 46340 by Daniel Poveda Parrilla, Aug 04 2016]

Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Formula

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

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

From Colin Barker, Aug 01 2016 and Aug 04 2016: (Start)

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

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

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

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).

(End)

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

E.g.f.: x*(2*(1 + 7*x + 6*x^2 + x^3)*exp(x) - exp(-x)). - G. C. Greubel, Aug 05 2016

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

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

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

Example

a(5) = 5^4 + Sum_{k=0..(5^2 - (5 mod 2))} 2k = 625 + Sum_{k=0..(25 - 1)} 2k = 625 + 600 = 1225.
a(12) = 12^4 + Sum_{k=0..(12^2 - (12 mod 2))} 2k = 20736 + Sum_{k=0..(144 - 0)} 2k = 20736 + 20880 = 41616.

Mathematica

Table[n^2 ((-1)^n + 2 n^2), {n, 0, 34}] (* or *)
CoefficientList[Series[x (1 + 34 x + 79 x^2 + 156 x^3 + 79 x^4 + 34 x^5 +
x^6)/((1 - x)^5 (1 + x)^3), {x, 0, 34}], x] (* Michael De Vlieger, Aug 01 2016 *)
LinearRecurrence[{2, 2, -6, 0, 6, -2, -2, 1}, {0, 1, 36, 153, 528, 1225, 2628, 4753}, 40] (* Harvey P. Dale, Sep 10 2016 *)

Prog

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

Crossrefs

Cf. A000040, A000129, A000217, A001110, A077221, A093954, A275543.

Sequence in context: A263120 A034592 A160754 * A117511 A250632 A211728

Adjacent sequences: A275493 A275494 A275495 * A275497 A275498 A275499

Keyword

nonn,easy

Author

Daniel Poveda Parrilla, Jul 30 2016

Extensions

New name from Colin Barker, Aug 04 2016

Status

approved