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A108563
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Number of representations of n as sum of twice a square plus thrice a square.
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0
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1, 0, 2, 2, 0, 4, 0, 0, 2, 0, 0, 4, 2, 0, 4, 0, 0, 0, 2, 0, 4, 4, 0, 0, 0, 0, 0, 2, 0, 4, 4, 0, 2, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 4, 4, 0, 0, 2, 0, 6, 0, 0, 4, 0, 0, 4, 0, 0, 4, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0, 0, 0, 2, 0, 0, 6, 0, 8, 0, 0, 4, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 6, 4, 0, 4
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Number of solutions to n = 2*a^2 + 3*b^2 in integers.
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).
a(n) > 0 if and only if n is in A002480. a(n) < 2 if and only if n is in A002481. - Michael Somos Mar 01 2011
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LINKS
| A. Berkovich and H. Yesilyurt, Ramanujan's identities and representation of integers by certain binary and quaternary quadratic forms
M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| G.f.: 1 + Sum_{k>0} x^k * (1 + x^(4*k)) * (1 + x^(6*k)) / (1 + x^(12*k)) - Sum_{k>0} kronecker( k, 3) * x^k * (1 - x^(2*k)) / (1 + x^(4*k)).
G.f.: Sum_{ i, j = - inf .. inf} x^(2*i^2 + 3*j^2). - Michael Somos Mar 01 2011
Expansion of phi(q^2) * phi(q^3) in powers of q where phi() is a Ramanujan theta function.
a(n) = A000377(n) - A115660(n).
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EXAMPLE
| 1 + 2*x^2 + 2*x^3 + 4*x^5 + 2*x^8 + 4*x^11 + 2*x^12 + 4*x^14 + 2*x^18 + ...
a(0) = 1 since 0 = 2*0^2 + 3*0^2, a(5) = 4 since 5 = 2*1^2 + 3*1^2 = 2*(-1)^2 + 3*1^2 = 2*1^2 + 3*(-1)^2 = 2*(-1^2) + 3*(-1)^2.
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PROG
| (PARI) for(n=0, 120, print1(if(n<1, n==0, qfrep([2, 0; 0, 3], n)[n]*2), ", "))
(PARI) {a(n) = if( n<0, 0, polcoeff( 1 + 2 * x * Ser( qfrep([ 2, 0; 0, 3], n)), n))} /* Michael Somos Mar 01 2011 */
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CROSSREFS
| Cf. A000377, A0002480, A002481, A115660.
Sequence in context: A143396 A090657 A167001 * A138476 A131381 A112080
Adjacent sequences: A108560 A108561 A108562 * A108564 A108565 A108566
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KEYWORD
| nonn
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AUTHOR
| Ralf Stephan, May 13 2007
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EXTENSIONS
| Edited by Charles R Greathouse IV (charles.greathouse(AT)case.edu), Oct 28 2009
Edited by N. J. A. Sloane, Mar 04 2011
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