A072070 -id:A072070 - 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

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

A072069 Number of integer solutions to the equation 2x^2+y^2+32z^2=m for an odd number m=2n-1.

Original entry on oeis.org

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

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 2 a(n) = A072068(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 + 6*x^5 + 4*x^6 + 4*x^9 + 4*x^10 + 2*x^13 + 8*x^14 + ... - _Michael Somos_, Dec 26 2019
G.f. = 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
+ ...

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=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

Cf. A006991, A003273, A072068, A072070, A072071, A080918.

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

{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)^2 * eta(x^32 + A)^5 / (eta(x + A)^2 * eta(x^4 + A)^3 * eta(x^16 + A)^2 * eta(x^64 + A)^2), n))}; /* Michael Somos, Dec 26 2019 */

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

Expansion of 2 * q^(1/2) * eta(q^2)^5 * eta(q^8)^2 * eta(q^32)^5 / (eta(q)^2 * eta(q^4)^3 * eta(q^16)^2 * eta(q^64)^2) in powers of q. - Michael Somos, Dec 26 2019

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

A080965 Expansion of eta(q^2)^12/(eta(q)^4eta(q^4)^5) in powers of q.

Original entry on oeis.org

1, 4, 2, -8, -4, 8, -8, -16, 6, 12, 8, -8, -8, 24, 0, -16, 12, 16, 10, -24, -8, 16, -24, -16, 8, 28, 8, -32, -16, 8, 0, -32, 6, 32, 16, -16, -12, 40, -24, -16, 24, 16, 16, -40, -8, 40, 0, -32, 24, 36, 10, -16, -24, 24, -32, -48, 0, 32, 24, -24, -16, 40, 0, -48, 12, 16, 16

Offset: 0

Michael Somos, Feb 28 2003

sign

Euler transform of period 4 sequence [4,-8,4,-3,...].

Alois P. Heinz, Table of n, a(n) for n = 0..10000

a(n)=A080964(4n)=2*A072071(4n)-A072070(4n).
A083703(n)=(-1)^n a(n). a(2n)=0 iff n in A004215 (checked up to n=343).
a(2n)=0 iff A005875(n)=0.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add([-3, 4, -8, 4]
      [1+irem(d, 4)]*d, d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 05 2015

a[n_] := a[n] = If[n==0, 1, Sum[DivisorSum[j, {-3, 4, -8, 4}[[1 + Mod[#, 4]]]*#&]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Nov 25 2015, after Alois P. Heinz *)

a(n)=local(X); if(n<0,0,X=x+x*O(x^n); polcoeff(eta(X)^-4*eta(X^2)^12*eta(X^4)^-5,n))

G.f.: Product_{n>0} (1-x^(2n))^12/((1-x^n)^4(1-x^(4n))^5).

A080964 Euler transform of period-16 sequence [2,-3,2,1,2,-3,2,-6,2,-3,2,1,2,-3,2,-3,...].

Original entry on oeis.org

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

Offset: 0

Michael Somos, Feb 28 2003

sign

Alois P. Heinz, Table of n, a(n) for n = 0..20000 (1501 terms from G. C. Greubel)

eta[q_] := q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[eta[q^2]^5 *eta[q^8]^7/(eta[q]^2*eta[q^4]^4*eta[q^16]^3), {q, 0, 60}], q]; Table[a[[n]], {n, 1, 70}] (* G. C. Greubel, Jul 02 2018 *)

a(n)=local(X); if(n<0,0,X=x+x*O(x^n); polcoeff(eta(X)^-2*eta(X^2)^5*eta(X^4)^-4*eta(X^8)^7*eta(X^16)^-3,n))

a(4*n+2) = a(4*n+3) = 0.
a(n) = 2*A072071(n) - A072070(n).
a(4*n) = A080965(n).
a(4*n+1) = 2*A080966(n).
Expansion of eta(q^2)^5*eta(q^8)^7/(eta(q)^2*eta(q^4)^4*eta(q^16)^3) in powers of q. - G. C. Greubel, Jul 02 2018

A080966 Expansion of theta_4(q^2) * theta_2(q)^2/(4*q^(1/2)) in powers of q.

Original entry on oeis.org

1, 2, -1, -2, 0, -4, -1, 2, -4, 2, 4, 2, 1, -2, 4, 2, 4, 0, -4, 0, -3, 4, -4, -4, 0, -2, 0, -6, 0, 2, -1, -4, 4, -4, -4, 8, 4, 6, 0, 2, -8, 0, 7, 2, 4, 2, 4, 0, 0, -6, 4, 0, -4, 0, 0, 0, 1, -6, -4, 4, -8, -2, -4, 4, 0, 2, -4, -6, 0, -2, 4, -8, 1, 2, 0, 0, 4, 4, 4, -2, 4, 6, 0, -2, 0, -4, -8, 10, 8, 8, -1, 4, 4, 2, -4, -4, -8, 6, 4, -6, 8, -6, 4, 4

Offset: 0

Michael Somos, Feb 28 2003

sign

The nonzero quadrisection of A248395.

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) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

			q + 2*q^5 - q^9 - 2*q^13 - 4*q^21 - q^25 + 2*q^29 - 4*q^33 + ...

G. C. Greubel, Table of n, a(n) for n = 0..5000
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

QP = QPochhammer; s = QP[q^2]^6/(QP[q]^2*QP[q^4]) + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 14 2015, adapted from PARI *)
QP := QPochhammer; a:=CoefficientList[Series[QP[q^2]^6/(QP[q]^2*QP[q^4]), {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jul 01 2018 *)

{a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)^6/eta(x+A)^2/eta(x^4+A), n))}

G.f.: Product_{k>0} (1+x^k)^2*(1-x^(2k))^3/(1+x^(2k)).
Expansion of f(-q^4)*f(q)^2 in powers of q where f(-q)=f(-q,-q^2) is a Ramanujan theta function.
Expansion of q^(-1/4)*eta(q^2)^6/(eta(q)^2*eta(q^4)) in powers of q.
Euler transform of period-4 sequence [2,-4,2,-3,...].
G.f.: Product_{k>0} (1-x^(2*k))^3*(1+x^k)^2/(1+x^(2*k)).
2*a(n) = A080964(4*n+1) = 2*A072071(4*n+1) - A072070(4*n+1).

cp's OEIS Frontend

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

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A006991 Primitive congruent numbers.

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

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A072069 Number of integer solutions to the equation 2x^2+y^2+32z^2=m for an odd number m=2n-1.

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

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A080965 Expansion of eta(q^2)^12/(eta(q)^4eta(q^4)^5) in powers of q.

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Maple

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A080964 Euler transform of period-16 sequence [2,-3,2,1,2,-3,2,-6,2,-3,2,1,2,-3,2,-3,...].

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