A072069 -id:A072069 - cp's OEIS frontend

A003273 Congruent numbers: positive integers k for which there exists a right triangle having area k and rational sides.

5, 6, 7, 13, 14, 15, 20, 21, 22, 23, 24, 28, 29, 30, 31, 34, 37, 38, 39, 41, 45, 46, 47, 52, 53, 54, 55, 56, 60, 61, 62, 63, 65, 69, 70, 71, 77, 78, 79, 80, 84, 85, 86, 87, 88, 92, 93, 94, 95, 96, 101, 102, 103, 109, 110, 111, 112, 116, 117, 118, 119, 120, 124, 125, 126

Offset: 1

N. J. A. Sloane

Positive integers k such that x^2 + k*y^2 = z^2 and x^2 - k*y^2 = t^2 have simultaneous integer solutions. In other words, k is the difference of an arithmetic progression of three rational squares: (t/y)^2, (x/y)^2, (z/y)^2. Values of k corresponding to y=1 (i.e., an arithmetic progression of three integer squares) form A256418.
Tunnell shows that if a number is squarefree and congruent, then the ratio of the number of solutions of a pair of equations is 2. If the Birch and Swinnerton-Dyer conjecture is assumed, then determining whether a squarefree number k is congruent requires counting the solutions to a pair of equations. For odd k, see A072068 and A072069; for even k see A072070 and A072071.
If a number k is congruent, there are an infinite number of right triangles having rational sides and area k. All congruent numbers can be obtained by multiplying a primitive congruent number A006991 by a positive square number A000290.
Conjectured asymptotics (based on random matrix theory) on p. 453 of Cohen's book. - Steven Finch, Apr 23 2009

			24 is congruent because 24 is the area of the right triangle with sides 6,8,10.
5 is congruent because 5 is the area of the right triangle with sides 3/2, 20/3, 41/6 (although not of any right triangle with integer sides -- see A073120). - _Jonathan Sondow_, Oct 04 2013

Alter, Ronald; Curtz, Thaddeus B.; Kubota, K. K. Remarks and results on congruent numbers. Proceedings of the Third Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1972), pp. 27-35. Florida Atlantic Univ., Boca Raton, Fla., 1972. MR0349554 (50 #2047)
H. Cohen, Number Theory. I, Tools and Diophantine Equations, Springer-Verlag, 2007, p. 454. [From Steven Finch, Apr 23 2009]
R. Cuculière, "Mille ans de chasse aux nombres congruents", in Pour la Science (French edition of 'Scientific American'), No. 7, 1987, pp. 14-18.
L. E. Dickson, History of the Theory of Numbers, Vol. 2, pp. 459-472, AMS Chelsea Pub. Providence RI 1999.
R. K. Guy, Unsolved Problems in Number Theory, D27.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Congruent numbers up to 10000; table of n, a(n) for n = 1..5742
R. Alter, Letter to N. J. A. Sloane, Sep 1975
R. Alter, The congruent number problem, Amer. Math. Monthly, 87 (1980), 43-45.
R. Alter and T. B. Curtz, A note on congruent numbers, Math. Comp., 28 (1974), 303-305 and 30 (1976), 198.
A. Alvarado and E. H. Goins, Arithmetic progressions on conic sections, arXiv:1210.6612 [math.NT], 2012. [From _Jonathan Sondow_, Oct 25 2012]
Estelle Basor and Bill Hart, A trillion triangles, American Institute of Mathematics,
E. Brown, Three Fermat Trails to Elliptic Curves, 5. Congruent Numbers and Elliptic Curves (pp 8-11/17)
Graeme Brown, The Congruent Number Problem, 2014.
Jasbir S. Chahal, Some remarks on rational right triangles, Expos. Math. (2024).
B. Cipra, Tallying the class of congruent numbers, ScienceNOW, Sep 23 2009.
Clay Mathematics Institute, The Birch and Swinnerton-Dyer Conjecture
Raiza Corpuz, Constructing congruent number elliptic curves using 2-descent, arXiv:2006.08113 [math.NT], 2020.
R. Cuculière, Mille ans de chasse aux nombres congruents, Séminaire de Philosophie et Mathématiques, 2, 1988, p. 1-17.
Department of Pure Maths., Univ. Sheffield, Pythagorean triples and the congruent number problem [Cached copy]
A. Dujella, A. S.Janfeda, and S. Salami, A Search for High Rank Congruent Number Elliptic Curves, JIS 12 (2009) 09.5.8.
E. V. Eikenberg, The Congruent Number Problem
David Goldberg, Triangle Sides for Congruent Numbers less than 10000, arXiv:2106.07373 [math.NT], 2021.
Lorenz Halbeisen and Norbert Hungerbühler, Congruent number elliptic curves with rank at least two, arXiv:1809.02037 [math.NT], 2018.
Lorenz Halbeisen and Norbert Hungerbühler, Congruent Number Elliptic Curves Related to Integral Solutions of m^2 = n^2 + nl + n^2, J. Int. Seq., Vol. 22 (2019), Article 19.3.1.
Alvaro Lozano-Robledo, My #MegaFavNumber: 224,403,517,704,336,969,924,557,513,090,674,863,160,948,472,041, video (2020) [discusses congruent numbers and a(157)]
W. F. Hammond, A Reading of Karl Rubin's SUMO Slides on Rational Right Triangles and Elliptic Curves
Bill Hart, A Trillion Triangles, American Institute of Mathematics.
T. Komatsu, Congruent numbers and continued fractions, Fib. Quart., 50 (2012), 222-226. - From _N. J. A. Sloane_, Mar 04 2013
S. Komoto, T. Watanabe and H. Wada, 42553 is a congruent number.
G. Kramarz, All congruent numbers less than 2000, Math. Annalen, 273 (1986), 337-340.
G. Kramarz, All congruent numbers less than 2000, Math. Annalen, 273 (1986), 337-340. [Annotated, corrected, scanned copy]
Allan J. MacLeod, The congruent number descent of Komotu, Watanabe and Wada, arXiv:2005.02615 [math.NT], 2020.
MathDL, Five Mathematicians Capture Record Number of Congruent Numbers
Fidel Ronquillo Nemenzo, All congruent numbers less than 40000, Proc. Japan Acad. Ser. A Math. Sci., Volume 74, Number 1 (1998), 29-31.
Karl Rubin, Right triangles and elliptic curves
W. A. Stein, Introduction to the Congruent Number Problem
W. A. Stein, The Congruent Number Problem
Ye Tian, Congruent Numbers and Heegner Points, arXiv:1210.8231 [math.NT], 2012.
J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334.
D. J. Wright, The Congruent Number Problem

Cf. A006991, A072068, A072069, A072070, A072071, A073120, A165564, A182429, A256418, A259680-A259687.

(* The following Mathematica code assumes the truth of the Birch and Swinnerton-Dyer conjecture and uses the list of primitive congruent numbers produced by the Mathematica code in A006991: *)
For[cLst={}; i=1, i<=Length[lst], i++, n=lst[[i]]; j=1; While[n j^2<=maxN, cLst=Union[cLst, {n j^2}]; j++ ]]; cLst

Guy gives a table up to 1000.
Edited by T. D. Noe, Jun 14 2002
Comments revised by Max Alekseyev, Nov 15 2008
Comment corrected by Jonathan Sondow, Oct 10 2013

A006991 Primitive congruent numbers.

Original entry on oeis.org

5, 6, 7, 13, 14, 15, 21, 22, 23, 29, 30, 31, 34, 37, 38, 39, 41, 46, 47, 53, 55, 61, 62, 65, 69, 70, 71, 77, 78, 79, 85, 86, 87, 93, 94, 95, 101, 102, 103, 109, 110, 111, 118, 119, 127, 133, 134, 137, 138, 141, 142, 143, 145, 149, 151, 154, 157, 158, 159

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

nonn

Squarefree terms of A003273.

Assuming the Birch and Swinnerton-Dyer conjecture, determining whether a number n is congruent requires counting the solutions to a pair of equations. For odd n, see A072068 and A072069; for even n see A072070 and A072071. The Mathematica program for this sequence uses variables defined in A072068, A072069, A072070, A072071. - T. D. Noe, Jun 13 2002

			6 is congruent because 6 is the area of the right triangle with sides 3,4,5. It is a primitive congruent number because it is squarefree.

Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See p. 155.
R. K. Guy, Unsolved Problems in Number Theory, D27.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Primitive congruent numbers up to 10000; table of n, a(n) for n = 1..3503
R. Alter and T. B. Curtz, A note on congruent numbers, Math. Comp., 28 (1974), 303-305 and 30 (1976), 198.
American Institute of Mathematics, A trillion triangles
Jose Aranda, C++ program
B. Cipra, Tallying the class of congruent numbers, ScienceNOW, Sep 23 2009.
Clay Mathematics Institute, The Birch and Swinnerton-Dyer Conjecture
Keith Conrad, The Congruent Number Problem, The Harvard College Mathematics Review, 2008.
Department of Pure Maths., Univ. Sheffield, Pythagorean triples and the congruent number problem
A. Dujella, A. S. Janfeda, and S. Salami, A Search for High Rank Congruent Number Elliptic Curves, JIS 12 (2009) 09.5.8.
Hisanori Mishima, 361 Congruent Numbers g: 1<=g<=999
Giovanni Resta, Congruent numbers Primitive congruent numbers up to 10^7.
Karl Rubin, Elliptic curves and right triangles
J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334.
Wikipedia, Congruent number
R. G. Wilson v, Letter to N. J. A. Sloane, Oct. 1993

Cf. A003273, A072068, A072069, A072070, A072071.

(* The following Mathematica code assumes the truth of the Birch and Swinnerton-Dyer conjecture and uses functions from A072068. *)
For[lst={}; n=1, n<=maxN, n++, If[SquareFreeQ[n], If[(EvenQ[n]&&soln3[[n/2]]==2soln4[[n/2]])|| (OddQ[n]&&soln1[[(n+1)/2]]==2soln2[[(n+1)/2]]), AppendTo[lst, n]]]]; lst
(* The following self-contained Mathematica code also assumes the truth of the Birch and Swinnerton-Dyer conjecture. *)
CongruentQ[n_] := Module[{x, y, z, ok=False}, (Which[! SquareFreeQ[n], Null[], MemberQ[{5, 6, 7}, Mod[n, 8]], ok = True, OddQ@n&&Length@Solve[x^2+2y^2+8z^2==n, {x, y, z}, Integers]==2Length@Solve[x^2+2y^2+32z^2==n, {x, y, z}, Integers], ok=True, EvenQ@n&&Length@Solve[x^2+4y^2+8z^2==n/2, {x, y, z}, Integers]==2Length@ Solve[x^2 + 4 y^2 + 32 z^2 == n/2, {x, y, z}, Integers], ok=True]; ok)]; Select[Range[200], CongruentQ] (* Frank M Jackson, Jun 06 2016 *)

More terms from T. D. Noe, Feb 26 2003

A248394 q-Expansion of the modular form of weight 3/2, g*theta(2) in Tunnell's notation (see Comments).

Original entry on oeis.org

0, 1, 0, 2, 0, 0, 0, 0, 0, 1, 0, -2, 0, 0, 0, 0, 0, -4, 0, -2, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 4, 0, -4, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 1, 0, 4, 0, 0, 0, 0, 0, 4, 0, 2, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 4, 0, -2

Offset: 0

N. J. A. Sloane, Oct 18 2014

sign

g = q*Product_{m=1..oo} (1-q^(8*m))*(1-q^(16*m)),
theta(t) = Sum_{n=-oo..oo} q^(t*n^2).
Although the OEIS does not normally include sequences in which every other term is zero, this one is important enough to warrant an exception.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334.
Eric Weisstein's World of Mathematics, Jacobi Theta Functions

The nonzero bisection is A034950, which has further information and references.
Used in A248397-A248406.
Cf. A000122 (theta_3(q)), A072068, A072069, A080917, A080918, A248395.

# This produces a list of the first 100 terms:
g:=q*mul((1-q^(8*m))*(1-q^(16*m)),m=1..30);
g:=series(g,q,100);
th:=t->series( add(q^(t*n^2),n=-50..50), q, 100);
series(g*th(2),q,100);
seriestolist(%);
# Alternative with https://oeis.org/transforms.txt and the Somos Euler transform in A034950:
p8 := [2,-3,2,-2,2,-3,2,-3] ;
L := [seq(op(p8),i=1..10)] ;
EULER(%) ;
[1,op(%)] ;
[0,op(AERATE(%,1))] ; # R. J. Mathar, Nov 11 2014

QP = QPochhammer; s = q*QP[q^8]*QP[q^16]*EllipticTheta[3, 0, q^2] + O[q]^80; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015 *)

From Seiichi Manyama, Sep 30 2018: (Start)
Let q = exp(Pi i t).
theta_3(q) = 1 + 2*q + 2*q^4 + 2*q^9 + 2*q^16 + ... .
G.f.: (theta_3(q) - theta_3(q^4))*(theta_3(q^32) - theta_3(q^8)/2)*theta_3(q^2).
a(2*n-1) = A080918(2*n-1) - A080917(2*n-1)/2 = A072069(n) - A072068(n)/2 for n > 0. (End)

A072068 Number of integer solutions to the equation 2x^2+y^2+8z^2=m for an odd number m=2n-1.

Original entry on oeis.org

2, 4, 0, 0, 10, 12, 0, 0, 16, 12, 0, 0, 10, 16, 0, 0, 16, 24, 0, 0, 32, 12, 0, 0, 18, 24, 0, 0, 16, 36, 0, 0, 32, 12, 0, 0, 16, 28, 0, 0, 34, 36, 0, 0, 48, 24, 0, 0, 16, 36, 0, 0, 32, 36, 0, 0, 32, 24, 0, 0, 26, 24, 0, 0

Offset: 1

T. D. Noe, Jun 13 2002

nonn

Related to primitive congruent numbers A006991.
Assuming the Birch and Swinnerton-Dyer conjecture, the odd number 2n-1 is a congruent number if it is squarefree and a(n) = 2*A072069(n).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			a(2) = 4 because (1,1,0), (-1,1,0), (1,-1,0) and (-1,-1,0) are solutions when m=3.
G.f. = 2*x + 4*x^2 + 10*x^5 + 12*x^6 + 16*x^9 + 12*x^10 + 10*x^13 + 16*x^14 + 16*x^17 + ...
G.f. = 2*q + 4*q^3 + 10*q^9 + 12*q^11 + 16*q^17 + 12*q^19 + 10*q^25 + 16*q^27 + ...

T. D. Noe, Table of n, a(n) for n=1..10000
Clay Mathematics Institute, The Birch and Swinnerton-Dyer Conjecture
Department of Pure Maths., Univ. Sheffield, Pythagorean triples and the congruent number problem
Karl Rubin, Elliptic curves and right triangles
J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334.

Cf. A006991, A003273, A072069, A072070, A072071, A080917.

maxN=128; soln1=Table[0, {maxN/2}]; xMax=Ceiling[Sqrt[maxN/2]]; yMax=Ceiling[Sqrt[maxN]]; zMax=Ceiling[Sqrt[maxN/8]]; Do[n=2x^2+y^2+8z^2; If[OddQ[n]&&nA072068 = CoefficientList[s, x] // Rest (* Jean-François Alcover, Feb 16 2015, after Michael Somos *)

{a(n) = my(A); n--; if( n<0, 0, A = x * O(x^n); polcoeff( 2 * eta(x^2 + A)^5 * eta(x^8 + A)^7 / (eta(x + A)^2 * eta(x^4 + A)^5 * eta(x^16 + A)^2), n))}; /* Michael Somos, Dec 26 2019 */

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

Expansion of 2 * q^(1/2) * eta(q^2)^5 * eta(q^8)^7 / (eta(q)^2 * eta(q^4)^5 * eta(q^16)^2) in powers of q. - Michael Somos, Feb 19 2015

A072070 Number of integer solutions to the equation 4x^2 + y^2 + 8z^2 = n.

Original entry on oeis.org

1, 2, 0, 0, 4, 4, 0, 0, 6, 6, 0, 0, 8, 12, 0, 0, 12, 8, 0, 0, 8, 8, 0, 0, 8, 14, 0, 0, 16, 4, 0, 0, 6, 16, 0, 0, 12, 20, 0, 0, 24, 8, 0, 0, 8, 20, 0, 0, 24, 18, 0, 0, 24, 12, 0, 0, 0, 16, 0, 0, 16, 20, 0, 0, 12, 8, 0, 0, 16, 16, 0, 0, 30, 32, 0, 0, 24, 16, 0, 0, 24, 18, 0, 0, 16, 24, 0, 0, 24, 16

Offset: 0

T. D. Noe, Jun 13 2002

nonn

Related to primitive congruent numbers A006991.
Assuming the Birch and Swinnerton-Dyer conjecture, the even number 2n is a congruent number if it is squarefree and a(n) = 2 A072071(n).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			a(4) = 4 because (1, 0, 0), (-1, 0, 0), (0, 2, 0) and (0, -2, 0) are solutions.
G.f. = 1 + 2*q + 4*q^4 + 4*q^5 + 6*q^8 + 6*q^9 + 8*q^12 + 12*q^13 + 12*q^16 + 8*q^17 + ...

J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334.

T. D. Noe, Table of n, a(n) for n = 0..10000
Clay Mathematics Institute, The Birch and Swinnerton-Dyer Conjecture
Department of Pure Maths., Univ. Sheffield, Pythagorean triples and the congruent number problem
Karl Rubin, Elliptic curves and right triangles
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A006991, A003273, A072068, A072069, A072071, A080917.

maxN=128; soln3=Table[0, {maxN/2}]; xMax=Ceiling[Sqrt[maxN/8]]; yMax=Ceiling[Sqrt[maxN/2]]; zMax=Ceiling[Sqrt[maxN/16]]; Do[n=4x^2+y^2+8z^2; If[n>0&&n<=maxN/2, s=8; If[x==0, s=s/2]; If[y==0, s=s/2]; If[z==0, s=s/2]; soln3[[n]]+=s], {x, 0, xMax}, {y, 0, yMax}, {z, 0, zMax}]
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^4] EllipticTheta[ 3, 0, q^8], {q, 0, n}]; (* Michael Somos, Jul 23 2018 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^-2 * eta(x^2 + A)^5 * eta(x^4 + A)^-4 * eta(x^8 + A)^3 * eta(x^16 + A)^3 * eta(x^32 + A)^-2, n))}; /* Michael Somos, Feb 11 2003 */

Expansion of phi(q) * phi(q^4) * phi(q^8) in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Jun 09 2012
Euler transform of period 32 sequence [2, -3, 2, 1, 2, -3, 2, -2, 2, -3, 2, 1, 2, -3, 2, -5, 2, -3, 2, 1, 2, -3, 2, -2, 2, -3, 2, 1, 2, -3, 2, -3, ...]. - Michael Somos, Feb 11 2003
a(4*n + 2) = a(4*n + 3) = 0. a(4*n) = A014455(n). - Michael Somos, Jun 08 2012
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 2^(7/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A080917. - Michael Somos, Jul 23 2018

A072071 Number of integer solutions to the equation 4x^2+y^2+32z^2=n.

Original entry on oeis.org

1, 2, 0, 0, 4, 4, 0, 0, 4, 2, 0, 0, 0, 4, 0, 0, 4, 4, 0, 0, 8, 0, 0, 0, 0, 6, 0, 0, 0, 4, 0, 0, 6, 4, 0, 0, 12, 12, 0, 0, 16, 8, 0, 0, 0, 12, 0, 0, 8, 10, 0, 0, 24, 4, 0, 0, 0, 12, 0, 0, 0, 12, 0, 0, 12, 8, 0, 0, 16, 8, 0, 0, 20, 12, 0, 0, 0, 8, 0, 0, 8, 6, 0, 0, 16, 16, 0, 0, 0, 4, 0, 0, 0, 8, 0, 0, 8

Offset: 0

T. D. Noe, Jun 13 2002

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Related to primitive congruent numbers A006991.
Assuming the Birch and Swinnerton-Dyer conjecture, the even number 2n is a congruent number if it is squarefree and 2 a(n) = A072070(n).

			a(4) = 4 because (1,0,0), (-1,0,0), (0,2,0) and (0,-2,0) are solutions.
1 + 2*x + 4*x^4 + 4*x^5 + 4*x^8 + 2*x^9 + 4*x^13 + 4*x^16 + 4*x^17 + 8*x^20 + ...

J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334.

T. D. Noe, Table of n, a(n) for n = 0..10000
Clay Mathematics Institute, The Birch and Swinnerton-Dyer Conjecture
Department of Pure Maths., Univ. Sheffield, Pythagorean triples and the congruent number problem
Karl Rubin, Elliptic curves and right triangles
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A006991, A003273, A072068, A072069, A072070.

J12[q_] := Sum[q^n^2, {n, -10, 10}]; CoefficientList[Series[J12[q]J12[q^4]J12[q^32], {q, 0, 100}], q]

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^-2 * eta(x^2 + A)^5 * eta(x^4 + A)^-4 * eta(x^8 + A)^5 * eta(x^16 + A)^-2 * eta(x^32 + A)^-2 * eta(x^64 + A)^5 * eta(x^128 + A)^-2, n))}

Expansion of phi(x) * phi(x^4) * phi(x^32) in powers of x where phi() is a Ramanujan theta function.

a(4*n + 2) = a(4*n + 3) = 0. - Michael Somos, Jun 08 2012

More terms from Vladeta Jovovic, Jun 16 2002

A034950 Expansion of eta(8z)eta(16z)theta_3(2z).

Original entry on oeis.org

1, 2, 0, 0, 1, -2, 0, 0, -4, -2, 0, 0, -3, 0, 0, 0, 4, -4, 0, 0, 0, 6, 0, 0, 1, 4, 0, 0, 4, 2, 0, 0, 0, -2, 0, 0, 4, -2, 0, 0, -3, 2, 0, 0, -4, -4, 0, 0, -4, 2, 0, 0, -8, -6, 0, 0, 8, -4, 0, 0, 1, -4, 0, 0, -4, 6, 0, 0, 0, 2, 0, 0, 0, -2, 0, 0, 4, 8, 0, 0, 0, 6, 0, 0, 5, -2, 0, 0, 4, -2, 0, 0, 8, 4, 0, 0, -4, -8, 0, 0, -4, 8, 0, 0, 4

Offset: 0

N. J. A. Sloane

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 + 2*x + x^4 - 2*x^5 - 4*x^8 - 2*x^9 - 3*x^12 + 4*x^16 - 4*x^17 + ...
G.f. = q + 2*q^3 + q^9 - 2*q^11 - 4*q^17 - 2*q^19 - 3*q^25 + 4*q^33 - ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
Ken Ono and Christopher Skinner, Fourier Coefficients of Half-Integral Weight Modular Forms Modulo l, Ann. Math., 147 (1998), 453-470.
Michael Somos, Introduction to Ramanujan theta functions
J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A072068, A072069, A080963.

A bisection of A248394.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] EllipticTheta[ 2, 0, x] EllipticTheta[ 2, Pi/4, x] / Sqrt[8 x], {x, 0, n}]; (* Michael Somos, Feb 18 2015 *)
QP = QPochhammer; s = QP[q^2]^5*(QP[q^8]/(QP[q]^2*QP[q^4])) + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^8 + A) / (eta(x + A)^2 * eta(x^4 + A)), n))}; /* Michael Somos, Feb 16 2006 */

Euler transform of period 8 sequence [2, -3, 2, -2, 2, -3, 2, -3, ...]. - Michael Somos, Feb 16 2006
Expansion of q^(-1/2) * eta(q^2)^5 * eta(q^8) / (eta(q)^2 * eta(q^4)) in powers of q. - Michael Somos, Feb 16 2006
Expansion of psi(x)^2 * psi(-x^2) = phi(x) * psi(x^2) * psi(-x^2) = phi(x) * psi(x^4) * phi(-x^4) in powers of x where phi(), psi() are Ramanujan theta functions. - Michael Somos, Feb 18 2015
G.f.: Product_{k>0} (1 + x^k)^2 * (1 - x^(2*k))^3 * (1 + x^(4*k)). - Michael Somos, Feb 16 2006
2 * a(n) = A080963(2*n + 1). a(4*n + 2) = a(4*n + 3) = 0. - Michael Somos, Feb 18 2015
a(n) = A072069(n+1) - A072068(n+1)/2. - Seichi Manymama, Sep 30 2018

cp's OEIS Frontend

A003273 Congruent numbers: positive integers k for which there exists a right triangle having area k and rational sides.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Extensions

A006991 Primitive congruent numbers.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Extensions

A248394 q-Expansion of the modular form of weight 3/2, g*theta(2) in Tunnell's notation (see Comments).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A072068 Number of integer solutions to the equation 2x^2+y^2+8z^2=m for an odd number m=2n-1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A072070 Number of integer solutions to the equation 4*x^2 + y^2 + 8*z^2 = n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A072071 Number of integer solutions to the equation 4x^2+y^2+32z^2=n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A034950 Expansion of eta(8z)*eta(16z)*theta_3(2z).

Views

Author

Keywords

Comments

A072070 Number of integer solutions to the equation 4x^2 + y^2 + 8z^2 = n.

A034950 Expansion of eta(8z)eta(16z)theta_3(2z).