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A143621 a(n) = (-1)^binomial(n,4): Periodic sequence 1,1,1,1,-1,-1,-1,-1,... . 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Periodic sequence with period 8. More generally, it appears that (-1)^binomial(n,2^r) gives a periodic sequence of period 2^(r+1), the period consisting of a block of 2^r plus ones followed by a block of 2^r minus ones. See A033999 (r = 0), A057077 (r = 1) and A143622 (r = 3).

Nonsimple continued fraction expansion of A188943 = 1.767591879243... - R. J. Mathar, Mar 08 2012

LINKS

Table of n, a(n) for n=0..84.

Maciej Gawron, and Maciej Ulas, On formal inverse of the Prouhet-Thue-Morse sequence, Discrete Mathematics 339.5 (2016): 1459-1470. Also arXiv preprintarXiv:1601.04840 [math.CO], 2016. The sequence appears on page 1464, prefixed by three 0's.

FORMULA

a(n) = (-1)^binomial(n,4) = (-1)^floor(n/4), since sum {k = 1..n-3} k(k+1)(k+2)/3! = binomial(n,4) == floor(n/4) (mod 2) for n = 0,1,...,7 by calculation and both sides increase by an even number if we substitute n+8 for n.

a(n) = 1/4*((n+4) mod 8 - n mod 8).

O.g.f.: (1+x+x^2+x^3)/(1+x^4) = (1+x)*(1+x^2)/(1+x^4) = (1-x^4)/((1-x)*(1+x^4)).

Define E(k) = sum {n = 0..inf} a(n)*n^k/n! for k = 0,1,2,... . Then E(k) is an integral linear combination of E(0), E(1), E(2) and E(3) (a Dobinski-type relation).

a(n) = (-1)^A180969(2,n), where the first index in A180969(.,.) is the row index. - Adriano Caroli, Nov 18 2010

Euler transform of length 8 sequence [ 1, 0, 0, -2, 0, 0, 0, 1]. - Michael Somos, Sep 30 2011

G.f.: (1 - x^4)^2 / ((1 - x) * (1 - x^8)). a(n) = -a(-1 - n) for all n in Z. - Michael Somos, Sep 30 2011

E.g.f.: sin(x/sqrt(2))*sinh(x/sqrt(2)) + (sqrt(2)*sin(x/sqrt(2)) + cos(x/sqrt(2)))*cosh(x/sqrt(2)). - Ilya Gutkovskiy, Apr 15 2016

EXAMPLE

G.f. = 1 + x + x^2 + x^3 - x^4 - x^5 - x^6 - x^7 + x^8 + x^9 + x^10 + ...

MAPLE

with(combinat):

a := n -> (-1)^binomial(n, 4):

seq(a(n), n = 0..103);

MATHEMATICA

Table[(-1)^Binomial[n, 4], {n, 0, 100}] (* Wesley Ivan Hurt, May 20 2014 *)

a[ n_] := (-1)^Quotient[n, 4]; (* Michael Somos, May 05 2015 *)

PROG

(PARI) {a(n) = (-1)^(n \ 4)}; /* Michael Somos, Sep 30 2011 */

(PARI) x='x+O('x^99); Vec((1-x^4)^2/((1-x)*(1-x^8))) \\ Altug Alkan, Apr 15 2016

CROSSREFS

Cf. A033999, A057077, A130151, A143622.

Sequence in context: A174351 A181432 A165326 * A292117 A098417 A143622

Adjacent sequences:  A143618 A143619 A143620 * A143622 A143623 A143624

KEYWORD

easy,sign,changed

AUTHOR

Peter Bala, Aug 30 2008

STATUS

approved

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Last modified January 19 06:37 EST 2020. Contains 331033 sequences. (Running on oeis4.)