A162395 a(n) = -(-1)^n * n^2.

1, -4, 9, -16, 25, -36, 49, -64, 81, -100, 121, -144, 169, -196, 225, -256, 289, -324, 361, -400, 441, -484, 529, -576, 625, -676, 729, -784, 841, -900, 961, -1024, 1089, -1156, 1225, -1296, 1369, -1444, 1521, -1600, 1681, -1764, 1849, -1936, 2025, -2116, 2209, -2304, 2401, -2500

Offset: 1

Author

Michael Somos, Jul 02 2009

Keywords

• sign
• mult
• easy

Comments

This sequence is the denominator of (Pi^2)/12 = 1/1-1/4+1/9-1/16+1/25-1/36+... - Mohammad K. Azarian, Dec 29 2011

Also, circulant determinant of [1,2,...,n,n-1,...,1], i.e., determinant of the (2n-1) X (2n-1) matrix which has this as first row (and also first column), where row k+1 is obtained by cyclically shifting row k one place to the left. - M. F. Hasler, Dec 17 2016

Examples

			G.f. = x - 4*x^2 + 9*x^3 - 16*x^4 + 25*x^5 - 36*x^6 + 49*x^7 - 64*x^8 + 81*x^9 + ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Michael Somos, Rational Function Multiplicative Coefficients.
• Index entries for linear recurrences with constant coefficients, signature (-3,-3,-1).

Crossrefs

Cf. A000290, A072691.

For the reversion of this sequence see A263843 (and also A007297).

Programs

Magma

[(-1)^(n+1) * n^2: n in [1..60]]; // Vincenzo Librandi, Feb 15 2013

Mathematica

Table[(-1)^(n+1) * n^2, {n, 60}] (* Vincenzo Librandi, Feb 15 2013 *)

PARI

{a(n) = -(-1)^n * n^2};

Formula

Euler transform of length 2 sequence [-4, 3].

a(n) is multiplicative with a(2^e) = -(4^e) if e>0, a(p^e) = (p^2)^e if p>2.

G.f.: x * (1 - x) / (1 + x)^3.

E.g.f.: exp(-x) * (x - x^2).

a(n) = a(-n) = -(-1)^n * A000290(n) for all n in Z.

Sum_{n>=1} 1/a(n) = Pi^2/12 (A072691). - Amiram Eldar, Dec 10 2022

Dirichlet g.f.: zeta(s-2)*(1-2^(3-s)) = DirichletEta(s-2). - Amiram Eldar, Jan 07 2023