OFFSET
1,2
COMMENTS
Let A be the Hessenberg n X n matrix defined by: A[1,j]=j mod 2, A[i,i]:=1, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=3, a(n-1)=(-1)^(n-1)*coeff(charpoly(A,x),x^(n-2)). - Milan Janjic, Jan 24 2010
Also triangular numbers with alternating signs. - Stanislav Sykora, Nov 26 2013
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (-3,-3,-1).
FORMULA
From R. J. Mathar, Nov 05 2011: (Start)
a(n) = Sum_{i=1..n} (-1)^i*i^2 = (-1)^n*n*(n+1)/2.
G.f.: -x / (1+x)^3. (End)
a(n) = (-1)^n*det(binomial(i+2,j+1), 1 <= i,j <= n-1). - Mircea Merca, Apr 06 2013
G.f.: -W(0)/(2+2*x), where W(k) = 1 + 1/( 1 - x*(k+2)/( x*(k+2) - (k+1)/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 19 2013
E.g.f.: (1/2)*x*(x-2)*exp(-x). - G. C. Greubel, Mar 31 2021
Sum_{n>=1} 1/a(n) = 2 - 4*log(2). - Amiram Eldar, Jan 31 2023
EXAMPLE
a(6) = 1 + 4 - 9 + 16 - 25 + 36 = 3 + 7 + 11 = 21.
MAPLE
seq(sum(binomial(n, m), m=1..2)-n^2, n=2..51); # Zerinvary Lajos, Jun 19 2008
A089594 := n -> (-1)^n*n*(n+1)/2; # Peter Luschny, Jul 08 2011
MATHEMATICA
nn = Range[50]; Accumulate[(-1)^nn*nn^2] (* Jayanta Basu, Jun 06 2013 *)
PROG
(PARI) for(i=1, 50, print1(", "sum(j=1, i, (-1)^j*j^2)))
(PARI) a(n)=(-1)^n*n*(n+1)/2 \\ Charles R Greathouse IV, Jul 08 2011
(Magma) [(-1)^n*n*(n+1)/2: n in [1..50]]; // Vincenzo Librandi, Nov 16 2011
(Sage) [(-1)^n*binomial(n+1, 2) for n in (1..50)] # G. C. Greubel, Mar 31 2021
CROSSREFS
Cf. A059841 (p=0,q=-1), A130472 (p=1,q=-1), this sequence (p=2,q=-1), A232599 (p=3,q=-1), A126646 (p=0,q=2), A036799 (p=1,q=2), A036800 (p=2,q=2), A036827 (p=3,q=2), A077925 (p=0,q=-2), A232600 (p=1,q=-2), A232601 (p=2,q=-2), A232602 (p=3,q=-2), A232603 (p=2,q=-1/2), A232604 (p=3,q=-1/2).
Cf. A000217.
Cf. A225144. [Bruno Berselli, Jun 06 2013]
KEYWORD
sign,easy
AUTHOR
Jon Perry, Dec 30 2003
STATUS
approved