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Search: id:A035187
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| A035187 |
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Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 5. |
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+0
6
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| 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 1, 0, 0, 2, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 2, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 1, 2, 0, 0, 0, 0
(list; graph; listen)
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OFFSET
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1,11
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COMMENT
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Let tau be the golden ratio (1+sqrt(5))/2; let zetaQ(tau)(s)=sum(1/(Z(tau):a)^s) the Dedekind zeta function where a runs through the nonzero ideals of Z(tau) and where (Z(tau):a) is the norm of a; then zetaQ(tau)(s)=sum(n>=1,a(n)/n^s)
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LINKS
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M. Baake, Algebra, Combinatorics and Number Theory
M. Baake and R. V. Moody, Similarity submodules and root systems in four dimensions, Canad. J. Math. 51 (1999), 1258-1276.
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FORMULA
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Sum(k=1, n, a(k)) is asymptotic to c*n where c=2*log(tau)/sqrt(5)
Multiplicative with a(5^e) = 1, a(p^e) = e+1 if p == 1, 4 (mod 5), a(p^e) = (1+(-1)^e)/2 if p == 2, 3 (mod 5). - Michael Somos Jun 06 2005
Moebius transform is period 5 sequence [1, -1, -1, 1, 0, ...]. - Michael Somos Oct 29 2005
q-series for -a(n): Sum_{n >= 1} (-1)^nq^(n(n+1)/2)(1-q)(1-q^2)...(1-q^(n-1))/((1-q^(n+1))(1-q^(n+2))...(1-q^(2n))). [From Jeremy Lovejoy (lovejoy(AT)liafa.jussieu.fr), Jun 12 2009]
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PROGRAM
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(PARI) a(n)=if(n<1, 0, direuler(p=2, n, 1/(1-X)/(1-kronecker(5, p)*X))[n]) /* Michael Somos Jun 06 2005 */
(PARI) {a(n)=local(A, p, e); if(n<1, 0, A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==5, 1, if((p%5==1)|(p%5==4), e+1, !(e%2))))))} /* Michael Somos Jun 06 2005 */
(PARI) a(n)=if(n<1, 0, sumdiv(n, d, kronecker(5, d)))
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CROSSREFS
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Cf. A031363 (for denominators).078428.
Sequence in context: A086014 A025437 A066032 this_sequence A033770 A101668 A141846
Adjacent sequences: A035184 A035185 A035186 this_sequence A035188 A035189 A035190
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KEYWORD
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nonn,mult
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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Additional comments from Benoit Cloitre, Dec 29, 2002
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