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A143622 a(n) = (-1)^binomial(n,8): Periodic sequence 1,1,1,1,1,1,1,1,-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, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Periodic sequence with period 16. 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 A143621 (r = 2).

Nonsimple continued fraction expansion of (47+sqrt(445))/42 = 1.62131007404.. - R. J. Mathar, Mar 08 2012

LINKS

Table of n, a(n) for n=0..85.

FORMULA

a(n) = (-1)^binomial(n,8) = (-1)^floor(n/8), since sum {k = 1..n-7} k(k+1)...(k+6)/7! = binomial(n,8) == floor(n/8) (mod 2) for n = 0,1,...,15 by calculation and both sides increase by an even number if we substitute n+16 for n. a(n) = 1/8*((n+8) mod 16 - n mod 16). O.g.f.: (1+x+x^2+x^3+x^4+x^5+x^6+x^7)/(1+x^8 ) = (1+x)*(1+x^2)*(1+x^4)/(1+x^8) = (1-x^8)/((1-x)*(1+x^8)).

Define E(k) = sum {n = 0..inf} a(n)*n^k/n! for k = 0,1,2,... . Then E(k) is a an integral linear combination of E(0),E(1),...,E(7) (a Dobinski-type relation).

a(n)=(-1)^A180969(3,n)

MAPLE

with(combinat):

a := n -> (-1)^binomial(n, 8):

seq(a(n), n = 0..95);

CROSSREFS

A033999, A057077, A130151, A143621.

Sequence in context: A143621 A292117 A098417 * A246016 A306638 A076479

Adjacent sequences:  A143619 A143620 A143621 * A143623 A143624 A143625

KEYWORD

easy,sign

AUTHOR

Peter Bala, Aug 30 2008

STATUS

approved

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Last modified January 21 22:47 EST 2020. Contains 331129 sequences. (Running on oeis4.)