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A028586
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Theta series of lattice with Gram matrix [2 1; 1 3].
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4
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1, 0, 2, 4, 0, 0, 0, 4, 2, 0, 2, 0, 4, 0, 0, 4, 0, 0, 6, 0, 0, 0, 0, 4, 0, 0, 0, 8, 4, 0, 0, 0, 2, 0, 0, 4, 0, 0, 0, 0, 2, 0, 8, 4, 0, 0, 0, 4, 4, 0, 2, 0, 0, 0, 0, 0, 0, 0, 4, 0, 4, 0, 0, 12, 0, 0, 0, 4, 0, 0, 0, 0, 6, 0, 0, 4, 0, 0, 0, 0, 0, 0, 4, 4, 0, 0, 0, 8, 0, 0, 6, 0, 4, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 4
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A10054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
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LINKS
| John Cannon, Table of n, a(n) for n = 0..10000
A. Berkovich and H. Yesilyurt, Ramanujan's identities and representation of integers by certain binary and quaternary quadratic forms, page 8 equation (3.18)
M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| G.f.: Sum_{n,m} x^(2*n^2 + 2*m*n + 3*m^2). - Michael Somos Jan 31 2011
Expansion of (theta_3(z)*theta_3(5z)+theta_2(z)*theta_2(5z)).
Expansion of phi(q^2) * phi(q^10) + 4 * q^3 * psi(q^4) * psi(q^20) in powers of q where phi(q),psi(q) are Ramanujan theta functions. - Michael Somos Aug 13 2006
If p is prime then a(p) is nonzero iff p is in A106865.
0=a(n)a(2n) and 2*A035170(n)=a(n)+a(2n) if n>0. - Michael Somos Oct 21 2006
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EXAMPLE
| 1 + 2*q^2 + 4*q^3 + 4*q^7 + 2*q^8 + 2*q^10 + 4*q^12 + 4*q^15 + 6*q^18 + 4*q^23 + 8*q^27 + 4*q^28 + 2*q^32 + 4*q^35 + 2*q^40 + 8*q^42 + 4*q^43 + 4*q^47 + ...
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PROG
| (PARI) {a(n) = if( n<1, n==0, qfrep([2, 1; 1, 3], n)[n] * 2)} /* Michael Somos Aug 13 2006 */
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CROSSREFS
| Sequence in context: A138758 A107501 A126732 * A072069 A004025 A102561
Adjacent sequences: A028583 A028584 A028585 * A028587 A028588 A028589
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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