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A084091 Expansion of sum(k>=0, x^2^k/(1+x^2^k+x^2^(k+1))). 7
0, 1, 0, 0, 1, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 1, -1, 0, 1, -1, 0, 0, -1, 0, 1, 0, 0, 1, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, -1, 0, 0, -1, 0, 1, 0, 0, 1, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 1, -1, 0, 1, -1, 0, 0, -1, 0, 1, 0, 0, 1, -1, 0, 1, -1, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, -1, 0, 0, -1, 0, 1, 0, 0, 1, -1, 0, 1, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Chances of values -1/0/+1 are ~2:5:2.

LINKS

Table of n, a(n) for n=0..104.

FORMULA

a(2n) = a(n) + 1 - (n+1 mod 3), a(2n+1) = 1 - (n mod 3). - Ralf Stephan, Sep 27 2003

a(n) is multiplicative with a(2^e) = (1 + (-1)^e)/2, a(3^e) = 0^e, a(p^e) = 1 if p == 1 (mod 6), a(p^e) = (-1)^e if p == 5 (mod 6). - Michael Somos, Jul 18 2004

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2 - v^2 + 2*w*(v-u) + w-v. - Michael Somos, Jul 18 2004

G.f.: Sum_{k>=0} f(x^2^k) where f(x) := x * (1 - x) / (1 - x^3). - Michael Somos, Jul 18 2004

max(sum(0<=k<=n,a(k))) = floor(log_4(n))+1. Proof by Nikolaus Meyberg.

Dirichlet g.f. (conjectured): L(chi_2(3),s)/(1-2^(-s)), with chi_2(3) the nontrivial Dirichlet character modulo 3. - Ralf Stephan, Mar 27 2015

a(2*n + 1) = A057078(n). a(3*n) = 0. a(3*n + 1) = A098725(n+1). - Michael Somos, Jun 16 2015

EXAMPLE

G.f. = x + x^4 - x^5 + x^7 - x^11 + x^13 + x^16 - x^17 + x^19 - x^20 - x^23 + ...

MATHEMATICA

a[ n_] := If[n < 1, 0, With[ {f = #/(1 + # + #^2) &}, SeriesCoefficient[ Sum[ f[x^2^k], {k, 0, Log[2, n]}], {x, 0, n}]]]; (* Michael Somos, Jun 16 2015 *)

PROG

(PARI) {a(n) = my(A, m); if( n<1, 0, A = O(x); m=1; while( m<=n, m*=2; A = x / (1 + x + x^2) + subst(A, x, x^2)); polcoeff(A, n))}; /* Michael Somos, Jul 18 2004 */

(PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, !(e%2), p==3, 0, kronecker( -12, p)^e)))}; /* Michael Somos, Jun 16 2015 */

(PARI) {a(n) = if( n<1, 0, direuler( p=1, n, if( p==2, 1 / (1 - X^2), p==3, 1, 1 / (1 - kronecker( -12, p) * X)))[n])}; /* Michael Somos, Jun 16 2015 */

CROSSREFS

Cf. A002487.

Positions of 0 are in A084090, of 1 in A084089, of -1 in A084088, of a(n)!=0 in A084087.

Cf. A057078, A098725.

Sequence in context: A244735 A245938 A176405 * A080846 A082401 A157238

Adjacent sequences: A084088 A084089 A084090 * A084092 A084093 A084094

KEYWORD

sign,mult

AUTHOR

Ralf Stephan, May 11 2003

STATUS

approved

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Last modified December 7 09:17 EST 2022. Contains 358654 sequences. (Running on oeis4.)