

A072069


Number of integer solutions to the equation 2x^2+y^2+32z^2=m for an odd number m=2n1.


9



2, 4, 0, 0, 6, 4, 0, 0, 4, 4, 0, 0, 2, 8, 0, 0, 12, 8, 0, 0, 16, 12, 0, 0, 10, 16, 0, 0, 12, 20, 0, 0, 16, 4, 0, 0, 12, 12, 0, 0, 14, 20, 0, 0, 20, 8, 0, 0, 4, 20, 0, 0, 8, 12, 0, 0, 24, 8, 0, 0, 14, 8, 0, 0
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OFFSET

1,1


COMMENTS

Related to primitive congruent numbers A006991.
Assuming the Birch and SwinnertonDyer conjecture, the odd number 2n1 is a congruent number if it is squarefree and 2 a(n) = A072068(n).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).


REFERENCES

J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323334.


LINKS

T. D. Noe, Table of n, a(n) for n=1..10000
Clay Mathematics Institute, The Birch and SwinnertonDyer Conjecture
Department of Pure Maths., Univ. Sheffield, Pythagorean triples and the congruent number problem
Karl Rubin, Elliptic curves and right triangles


FORMULA

Expansion of 2 * x * phi(x) * psi(x^4) * phi(x^16) in powers of x where phi(), psi() are Ramanujan theta functions.  Michael Somos, Jun 08 2012


EXAMPLE

a(2) = 4 because (1,1,0), (1,1,0), (1,1,0) and (1,1,0) are solutions when m=3.
2*q + 4*q^3 + 6*q^9 + 4*q^11 + 4*q^17 + 4*q^19 + 2*q^25 + 8*q^27 + 12*q^33
+ ...


MATHEMATICA

maxN=128; soln2=Table[0, {maxN/2}]; xMax=Ceiling[Sqrt[maxN/2]]; yMax=Ceiling[Sqrt[maxN]]; zMax=Ceiling[Sqrt[maxN/32]]; Do[n=2x^2+y^2+32z^2; If[OddQ[n]&&n<maxN, s=8; If[x==0, s=s/2]; If[y==0, s=s/2]; If[z==0, s=s/2]; soln2[[(n+1)/2]]+=s], {x, 0, xMax}, {y, 0, yMax}, {z, 0, zMax}]


CROSSREFS

Cf. A006991, A003273, A072068, A072070, A072071, A080918.
Sequence in context: A292144 A300324 A298368 * A230423 A213672 A309244
Adjacent sequences: A072066 A072067 A072068 * A072070 A072071 A072072


KEYWORD

nonn


AUTHOR

T. D. Noe, Jun 13 2002


STATUS

approved



