OFFSET
1,2
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
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).
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)). - Amiram Eldar, Jan 07 2023
EXAMPLE
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 + ...
MATHEMATICA
Table[(-1)^(n+1) * n^2, {n, 60}] (* Vincenzo Librandi, Feb 15 2013 *)
PROG
(PARI) {a(n) = -(-1)^n * n^2};
(Magma) [(-1)^(n+1) * n^2: n in [1..60]]; // Vincenzo Librandi, Feb 15 2013
CROSSREFS
KEYWORD
sign,mult,easy
AUTHOR
Michael Somos, Jul 02 2009
STATUS
approved