A087960 a(n) = (-1)^binomial(n+1,2).

1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1

Offset

0,1

Comments

Period 4: repeat [1, -1, -1, 1]. - Joerg Arndt, Feb 14 2016

Also equal to the sign of product(j-i, 1<=j<i<=n+1) = the sign of the Vandermonde determinant for -1, -2, ..., -(n+1).

Hankel transform of A097331, A097332. [Paul Barry, Aug 10 2009]

The Kn22 sums, see A180662, of triangle A108299 equal the terms of this sequence. [Johannes W. Meijer, Aug 14 2011]

References

I. S. Gradstein and I. M. Ryshik, Tables of series, products, and integrals, Volume 1, Verlag Harri Deutsch, 1981.

Links

Table of n, a(n) for n=0..100.

Gary Detlefs and Wolfdieter Lang, Improved Formula for the Multi-Section of the Linear Three-Term Recurrence Sequence, arXiv:2304.12937 [math.CO], 2023.

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

Formula

a(n) = (-1)^A000217(n).

a(n) = (-1)^floor((n+1)/2). - Benoit Cloitre and Ray Chandler, Sep 19 2003

G.f.: (1-x)/(1+x^2). - Paul Barry, Aug 10 2009

a(n) = I^(n*(n+1)). - Bruno Berselli, Oct 17 2011

a(n) = Product_{k=1..n} 2*cos(2*k*Pi/(2*n+1)) for n>=0 (for n=0 the empty product is put to 1). See the Gradstein-Ryshik reference, p. 63, 1.396 2. with x = sqrt(-1). - Wolfdieter Lang, Oct 22 2013

a(n) + a(n-2) = 0 for n>1, a(n) = a(n-4) for n>3. - Wesley Ivan Hurt, Jul 07 2016

E.g.f.: cos(x) - sin(x). - Ilya Gutkovskiy, Jul 07 2016

a(n) = Sum_{s=0..n} (-1)^(n-s)*A111125(n, s)*2^s (row polynomials of signed A111125 evaluated at 2). - Wolfdieter Lang, May 02 2021

Example

a(1) = -1 since (-1)^binomial(2,2) = (-1)^1 = -1.
G.f. = 1 - x - x^2 + x^3 + x^4 - x^5 - x^6 + x^7 + x^8 - x^9 - x^10 + ...

Maple

A087960:=n->(-1)^binomial(n+1, 2): seq(A087960(n), n=0..100); # Wesley Ivan Hurt, Jul 07 2016

Mathematica

(-1)^Binomial[Range[0, 110], 2] (* or *) LinearRecurrence[{0, -1}, {1, 1}, 110] (* Harvey P. Dale, Jul 07 2014 *)
a[ n_] := (-1)^(n (n + 1) / 2); (* Michael Somos, Jul 20 2015 *)
a[ n_] := (-1)^Quotient[ n + 1, 2]; (* Michael Somos, Jul 20 2015 *)

Prog

(PARI) {a(n) = (-1)^((n + 1)\2)}; /* Michael Somos, Jul 20 2015 */
(Haskell)
a087960 n = (-1) ^ (n * (n + 1) `div` 2)
a087960_list = cycle [1, -1, -1, 1]  -- Reinhard Zumkeller, Nov 15 2015
(Magma) [(-1)^Binomial(n+1, 2) : n in [0..100]]; // Wesley Ivan Hurt, Jul 07 2016
(Python)
def A087960(n): return -1 if n+1&2 else 1 # Chai Wah Wu, Jan 31 2023

Crossrefs

Cf. A000217, A021913, A057077, A097331, A097332, A108299, A111125, A180662.

Sequence in context: A265643 A283131 A008836 * A164660 A212159 A106400

Adjacent sequences: A087957 A087958 A087959 * A087961 A087962 A087963

Keyword

sign,easy

Author

W. Edwin Clark, Sep 17 2003

Extensions

More terms from Benoit Cloitre and Ray Chandler, Sep 19 2003

Offset and Vandermonde formula corrected by R. J. Mathar, Sep 25 2009

Status

approved