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Search: id:A133827
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| A133827 |
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Number of solutions to x + 7 * y = 2 * n in triangular numbers. |
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+0
1
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| 1, 0, 0, 1, 1, 2, 0, 0, 0, 0, 0, 2, 1, 0, 2, 0, 0, 0, 2, 0, 0, 2, 0, 0, 1, 0, 2, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 2, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 2, 0, 2, 0, 0, 0, 3, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 2, 2, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 2, 2, 0, 2, 0, 0, 2, 0, 2, 0
(list; graph; listen)
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OFFSET
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0,6
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COMMENT
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G.f. called omega(q) by Berkovich and Yesilyurt.
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LINKS
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A. Berkovich and H. Yesilyurt, New Identities For 7-cores with Prescribed BG-Rank, page 3 equation (1.19)
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FORMULA
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Expansion of phi(q) * psi(q^7) - q * psi(q^2) * psi(q^14) in powers of q^2 where psi() is a Ramanujan theta function.
Expansion of psi(q^4) * phi(q^14) + q^3 * psi(q^28) * phi(q^2) in powers of q where phi(), psi() are Ramanujan theta functions.
a(n) = b(2*n+1) where b(n) is multiplicative and b(2^e) = 0^e, b(7^e) = 1, b(p^e) = (1 + (-1)^e) / 2 if p == 3, 5, 6 (mod 7), b(p^e) = e + 1 if p == 1, 2, 4 (mod 7).
a(7*n+1) = a(7*n+2) = a(7*n+6) = 0. a(7*n+3) = a(n).
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EXAMPLE
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q + q^7 + q^9 + 2*q^11 + 2*q^23 + q^25 + 2*q^29 + 2*q^37 + 2*q^43 + q^49 + ...
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PROGRAM
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(PARI) {a(n) = if( n<0, 0, n = 2*n+1; sumdiv(n, d, (d%2) * kronecker( -28, d)))}
(PARI) {a(n) = local(A, p, e); if( n<0, 0, n = 2*n+1; A = factor(n); prod(k = 1, matsize(A) [1], if(p = A[k, 1], e = A[k, 2]; if(p == 2, 0, if( p == 7, 1, if( 1 == kronecker( -7, p), e + 1, !(e%2)) )))))}
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CROSSREFS
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A035162(2*n+1) = a(n).
Sequence in context: A065205 A036272 A083339 this_sequence A028633 A074039 A047764
Adjacent sequences: A133824 A133825 A133826 this_sequence A133828 A133829 A133830
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KEYWORD
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nonn
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AUTHOR
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Michael Somos, Sep 25 2007
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