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

Author

W. Edwin Clark, Sep 17 2003

Keywords

• sign
• easy

Comments

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

Also equal to the sign of product(j-i, 1<=j

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]

Examples

			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 + ...
		

References

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

Links

• 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).

Crossrefs

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

Programs

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

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 *)

PARI

{a(n) = (-1)^((n + 1)\2)}; /* Michael Somos, Jul 20 2015 */

Python

def A087960(n): return -1 if n+1&2 else 1 # Chai Wah Wu, Jan 31 2023

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

Extensions

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

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

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