A000700 -id:A000700 - cp's OEIS frontend

A004018 Theta series of square lattice (or number of ways of writing n as a sum of 2 squares). Often denoted by r(n) or r_2(n).

1, 4, 4, 0, 4, 8, 0, 0, 4, 4, 8, 0, 0, 8, 0, 0, 4, 8, 4, 0, 8, 0, 0, 0, 0, 12, 8, 0, 0, 8, 0, 0, 4, 0, 8, 0, 4, 8, 0, 0, 8, 8, 0, 0, 0, 8, 0, 0, 0, 4, 12, 0, 8, 8, 0, 0, 0, 0, 8, 0, 0, 8, 0, 0, 4, 16, 0, 0, 8, 0, 0, 0, 4, 8, 8, 0, 0, 0, 0, 0, 8, 4, 8, 0, 0, 16, 0, 0, 0, 8, 8, 0, 0, 0, 0, 0, 0, 8, 4, 0, 12, 8

Offset: 0

N. J. A. Sloane

Number of points in square lattice on the circle of radius sqrt(n). Equivalently, number of Gaussian integers of norm n (cf. Conway-Sloane, p. 106).
Let b(n)=A004403(n), then Sum_{k=1..n} a(k)*b(n-k) = 1. - John W. Layman
Theta series of D_2 lattice.
Number 6 of the 74 eta-quotients listed in Table I of Martin (1996).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
The zeros in this sequence correspond to those integers with an equal number of 4k+1 and 4k+3 divisors, or equivalently to those that have at least one 4k+3 prime factor with an odd exponent (A022544). - Ant King, Mar 12 2013
If A(q) = 1 + 4*q + 4*q^2 + 4*q^4 + 8*q^5 + ... denotes the o.g.f. of this sequence then the function F(q) := 1/4*(A(q^2) - A(q^4)) = ( Sum_{n >= 0} q^(2*n+1)^2 )^2 is the o.g.f. for counting the ways a positive integer n can be written as the sum of two positive odd squares. - Peter Bala, Dec 13 2013
Expansion coefficients of (2/Pi)*K, with the real quarter period K of elliptic functions, as series of the Jacobi nome q, due to (2/Pi)*K = theta_3(0,q)^2. See, e.g., Whittaker-Watson, p. 486. - Wolfdieter Lang, Jul 15 2016
Sum_{k=0..n} a(n) = A057655(n). Robert G. Wilson v, Dec 22 2016
Limit_{n->oo} (a(n)/n - Pi*log(n)) = A062089: Sierpinski's constant. - Robert G. Wilson v, Dec 22 2016
The mean value of a(n) is Pi, see A057655 for more details. - M. F. Hasler, Mar 20 2017

			G.f. = 1 + 4*q + 4*q^2 + 4*q^4 + 8*q^5 + 4*q^8 + 4*q^9 + 8*q^10 + 8*q^13 + 4*q^16 + 8*q^17 + 4*q^18 + 8*q^20 + 12*q^25 + 8*q^26 + ... . - _John Cannon_, Dec 30 2006

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 162, #16 (7), r(n).
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 106.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.23).
E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 15, p. 32, Lemma 2 (with the proof), p. 116, (9.10) first formula.
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 133.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 240, r(n).
W. König and J. Sprekels, Karl Weierstraß (1815-1897), Springer Spektrum, Wiesbaden, 2016, p. 186-187 and p. 280-281.
C. D. Olds, A. Lax and G. P. Davidoff, The Geometry of Numbers, Math. Assoc. Amer., 2000, p. 51.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 244-245.
E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, fourth edition, reprinted, 1958, Cambridge at the University Press.

T. D. Noe, Table of n, a(n) for n = 0..10000
G. E. Andrews, R. Lewis and Z.-G. Liu, An identity relating a theta series to a sum of Lambert series, Bull. London Math. Soc., 33 (2001), 25-31.
H. H. Chan and C. Krattenthaler, Recent progress in the study of representations of integers as sums of squares, arXiv:math/0407061 [math.NT], 2004.
S. Cooper and M. Hirschhorn, A combinatorial proof of a result from number theory, Integers 4 (2004), #A09.
Michael Gilleland, Some Self-Similar Integer Sequences
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62. (This sequence is called rho, see page 6)
M. D. Hirschhorn, Jacobi's Two-Square Theorem and Related Identities
M. D. Hirschhorn, Arithmetic Consequences of Jacobi's Two-Squares Theorem
M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
Jacobi-Legendre letters, Correspondance mathématique entre Legendre et Jacobi, J. Reine Angew. Math. 80 (1875) 205-279, letter of September 9, 1828, p. 240-243, formula for 2K/Pi on p. 242.
Christian Kassel and Christophe Reutenauer, The zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1505.07229v3 [math.AG], 2015. [Note that a later version of this paper has a different title and different contents, and the number-theoretical part of the paper was moved to the publication which is next in this list.]
Christian Kassel and Christophe Reutenauer, Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1610.07793 [math.NT], 2016.
Christian Kassel, Christophe Reutenauer, The Fourier expansion of eta(z)eta(2z)eta(3z)/eta(6z), arXiv:1603.06357 [math.NT], 2016.
M. Kontsevich and D. Zagier, Periods, Institut des Hautes Etudes Scientifiques 2001 IHES/M/01/22. Published in B. Engquist and W. Schmid, editors, Mathematics Unlimited - 2001 and Beyond, 2 vols., Springer-Verlag, 2001, pp. 771-808, section 2.3. Example 3.
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
G. Nebe and N. J. A. Sloane, Home page for this lattice
F. Richman, Counting Gaussian integers in a disk
Grant Sanderson, Pi hiding in prime regularities, 3Blue1Brown video (2017).
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
Michael Somos, Introduction to Ramanujan theta functions
G. Villemin, Sommes de 2 carrés
Eric Weisstein's World of Mathematics, Barnes-Wall Lattice
Eric Weisstein's World of Mathematics, Moebius Transform
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Sum of Squares Function
Eric Weisstein's World of Mathematics, Theta Series
Zach Wissner-Gross a.k.a. The Riddler, Riddler Express: Can You Climb Your Way To Victory?, FiveThirtyEight, Sep 24 2021.
G. Xiao, Two squares
Index entries for sequences related to sums of squares
Index entries for sequences mentioned by Glaisher
Index entries for "core" sequences

Row d=2 of A122141 and of A319574, 2nd column of A286815.
Partial sums - 1 give A014198.
A071385 gives records; A071383 gives where records occur.
Cf. A001481, A004020, A005883, A057655 (partial sums), A057961, A057962, A002654,
A104271, A105673.

# JacobiTheta3 is defined in A000122.
A004018List(len) = JacobiTheta3(len, 2)
A004018List(102) |> println # Peter Luschny, Mar 12 2018

Basis( ModularForms( Gamma1(4), 1), 100) [1]; /* Michael Somos, Jun 10 2014 */

(sum(x^(m^2),m=-10..10))^2;
# Alternative:
A004018list := proc(len) series(JacobiTheta3(0, x)^2, x, len+1);
seq(coeff(%, x, j), j=0..len-1) end:
t1 := A004018list(102);
r2 := n -> t1[n+1]; # Peter Luschny, Oct 02 2018

SquaresR[2,Range[0,110]] (* Harvey P. Dale, Oct 10 2011 *)
a[ n_] := SquaresR[ 2, n]; (* Michael Somos, Nov 15 2011 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^2, {q, 0, n}]; (* Michael Somos, Nov 15 2011 *)
a[ n_] := With[{m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ EllipticK[ m] / (Pi/2), {q, 0, n}]]; (* Michael Somos, Jun 10 2014 *)
a[ n_] := If[ n < 1, Boole[n == 0], 4 Sum[ KroneckerSymbol[-4, d], {d, Divisors@n}]]; (* or *) a[ n_] := SeriesCoefficient[ QPochhammer[ q^2]^10/(QPochhammer[ q] QPochhammer[ q^4])^4, {q, 0, n}]; (* Michael Somos, May 17 2015 *)

{a(n) = polcoeff( 1 + 4 * sum( k=1, n, x^k / (1 + x^(2*k)), x * O(x^n)), n)}; /* Michael Somos, Mar 14 2003 */

{a(n) = if( n<1, n==0, 4 * sumdiv( n, d, (d%4==1) - (d%4==3)))}; /* Michael Somos, Jul 19 2004 */

{a(n) = if( n<1, n==0, 2 * qfrep([ 1, 0; 0, 1], n)[n])}; /* Michael Somos, May 13 2005 */

a(n)=if(n==0,return(1)); my(f=factor(n)); 4*prod(i=1,#f~, if(f[i,1]%4==1, f[i,2]+1, if(f[i,2]%2 && f[i,1]>2, 0, 1))) \\ Charles R Greathouse IV, Sep 02 2015

from sympy import factorint
def a(n):
    if n == 0: return 1
    an = 4
    for pi, ei in factorint(n).items():
       if pi%4 == 1: an *= ei+1
       elif pi%4 == 3 and ei%2: return 0
    return an
print([a(n) for n in range(102)]) # Michael S. Branicky, Sep 24 2021

from math import prod
from sympy import factorint
def A004018(n): return prod(1 if p==2 else (e+1 if p&3==1 else (e+1)&1) for p, e in factorint(n).items())<<2 if n else 1 # Chai Wah Wu, Jul 07 2022, corrected Jun 21 2024.

Q = DiagonalQuadraticForm(ZZ, [1]*2)
Q.representation_number_list(102) # Peter Luschny, Jun 20 2014

Expansion of theta_3(q)^2 = (Sum_{n=-oo..+oo} q^(n^2))^2 = Product_{m>=1} (1-q^(2*m))^2 * (1+q^(2*m-1))^4; convolution square of A000122.
Factor n as n = p1^a1 * p2^a2 * ... * q1^b1 * q2^b2 * ... * 2^c, where the p's are primes == 1 (mod 4) and the q's are primes == 3 (mod 4). Then a(n) = 0 if any b is odd, otherwise a(n) = 4*(1 + a1)*(1 + a2)*...
G.f. = s(2)^10/(s(1)^4*s(4)^4), where s(k) := subs(q=q^k, eta(q)) and eta(q) is Dedekind's function, cf. A010815. [Fine]
a(n) = 4*A002654(n), n > 0.
Expansion of eta(q^2)^10 / (eta(q) * eta(q^4))^4 in powers of q. - Michael Somos, Jul 19 2004
Expansion of ( phi(q)^2 + phi(-q)^2 ) / 2 in powers of q^2 where phi() is a Ramanujan theta function.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (u - v)^2 - (v - w) * 4 * w. - Michael Somos, Jul 19 2004
Euler transform of period 4 sequence [4, -6, 4, -2, ...]. - Michael Somos, Jul 19 2004
Moebius transform is period 4 sequence [4, 0, -4, 0, ...]. - Michael Somos, Sep 17 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 2 (t/i) f(t) where q = exp(2 Pi i t).
The constant sqrt(Pi)/Gamma(3/4)^2 produces the first 324 terms of the sequence when expanded in base exp(Pi), 450 digits of the constant are necessary. - Simon Plouffe, Mar 03 2011
a(n) = A004531(4*n). a(n) = 2*A105673(n), if n>0.
Let s = 16*q*(E1*E4^2/E2^3)^8 where Ek = Product_{n>=1} (1-q^(k*n)) (s=k^2 where k is elliptic k), then the g.f. is hypergeom([+1/2, +1/2], [+1], s) (expansion of 2/Pi*ellipticK(k) in powers of q). - Joerg Arndt, Aug 15 2011
Dirichlet g.f. Sum_{n>=1} a(n)/n^s = 4*zeta(s)*L_(-4)(s), where L is the D.g.f. of the (shifted) A056594. [Raman. J. 7 (2003) 95-127]. - R. J. Mathar, Jul 02 2012
a(n) = floor(1/(n+1)) + 4*floor(cos(Pi*sqrt(n))^2) - 4*floor(cos(Pi*sqrt(n/2))^2) + 8*Sum_{i=1..floor(n/2)} floor(cos(Pi*sqrt(i))^2)*floor(cos(Pi*sqrt(n-i))^2). - Wesley Ivan Hurt, Jan 09 2013
From Wolfdieter Lang, Aug 01 2016: (Start)
A Jacobi identity: theta_3(0, q)^2 = 1 + 4*Sum_{r>=0} (-1)^r*q^(2*r+1)/(1 - q^(2*r+1)). See, e.g., the Grosswald reference (p. 15, p. 116, but p. 32, Lemma 2 with the proof, has the typo r >= 1 instead of r >= 0 in the sum, also in the proof). See the link with the Jacobi-Legendre letter.
Identity used by Weierstraß (see the König-Sprekels book, p. 187, eq. (5.12) and p. 281, with references, but there F(x) from (5.11) on p. 186 should start with nu =1 not 0): theta_3(0, q)^2 = 1 + 4*Sum_{n>=1} q^n/(1 + q^(2*n)). Proof: similar to the one of the preceding Jacobi identity. (End)
a(n) = (4/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - Seiichi Manyama, May 27 2017
G.f.: Theta_3(q)^2 = hypergeometric([1/2, 1/2],[1],lambda(q)), with lambda(q) = Sum_{j>=1} A115977(j)*q^j. See the Kontsevich and Zagier link, with Theta -> Theta_3, z -> 2*z and q -> q^2. - Wolfdieter Lang, May 27 2018

A081362 Expansion of q^(1/24) * eta(q) / eta(q^2) in powers of q.

Original entry on oeis.org

1, -1, 0, -1, 1, -1, 1, -1, 2, -2, 2, -2, 3, -3, 3, -4, 5, -5, 5, -6, 7, -8, 8, -9, 11, -12, 12, -14, 16, -17, 18, -20, 23, -25, 26, -29, 33, -35, 37, -41, 46, -49, 52, -57, 63, -68, 72, -78, 87, -93, 98, -107, 117, -125, 133, -144, 157, -168, 178, -192, 209, -223, 236, -255, 276, -294, 312, -335, 361, -385

Offset: 0

Michael Somos, Mar 18 2003

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
(Number of partitions of n into an even number of parts) - (number of partitions of n into an odd number of parts). [Fine]
Number 3 of the 130 identities listed in Slater 1952. - Michael Somos, Aug 20 2015

			G.f. = 1 - x - x^3 + x^4 - x^5 + x^6 - x^7 + 2*x^8 - 2*x^9 + 2*x^10 - 2*x^11 + ...
G.f. = 1/q - q^23 - q^71 + q^95 - q^119 + q^143 - q^167 + 2*q^191 - 2*q^215 + ...

B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 425, Corollary 1, Eqs. (37)-(40).
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 38, Eq. (22.14).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..5000 from Vaclav Kotesovec)
Blecher, Aubrey; Brennan, Charlotte; Knopfmacher, Arnold; Mansour, Toufik Counting corners in partitions. Ramanujan J. 39, No. 1, 201-224 (2016).
J. Fulman, Random matrix theory over finite fields, Bull. Amer. Math. Soc. (N.S.), 39 (2002), no. 1, 51--85. MR1864086 (2002i:60012). See top of page 70, Eq. 3, with k=0. - _N. J. A. Sloane_, Aug 31 2014
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 13.
Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, function p_{e-o}(n).
Lucy Joan Slater, Further Identities of the Rogers-Ramanujan Type, Proc. London Math. Soc., Series 2, vol.s2-54, no.2, pp.147-167, (1952).
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000009, A000700, A069910, A069911, A072233.

a[ n_] := SeriesCoefficient[ Product[1 - x^k, {k, 1, n, 2}], {x, 0, n}];
a[ n_] := With[ {t = Log[q] / (2 Pi I)}, SeriesCoefficient[ q^(1/24) DedekindEta[t] / DedekindEta[2 t], {q, 0, n}]];
a[ n_] := SeriesCoefficient[ QHypergeometricPFQ[ {}, {}, x^2, x], {x, 0, n}]; (* Michael Somos, Jan 02 2015 *)
a[ n_] := SeriesCoefficient[ 1 / QPochhammer[ -x, x], {x, 0, n}]; (* Michael Somos, Jan 02 2015 *)
a[ n_] := SeriesCoefficient[ 1 / Product[ 1 + x^k, {k, n}], {x, 0, n}]; (* Michael Somos, Jan 02 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2], {x, 0, n}]; (* Michael Somos, Jan 02 2015 *)
a[ n_] := SeriesCoefficient[ QHypergeometricPFQ[ {}, {-1}, x, -1] 2, {x, 0, n}]; (* Michael Somos, May 11 2015 *)
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (m / ( 1 - m)^2 / (16 q))^(-1/24), {q, 0, n}]]; (* Michael Somos, May 11 2015 *)

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

from sage.combinat.partition import number_of_partitions_length
print([sum((-1)^k*number_of_partitions_length(n, k) for k in (0..n)) for n in (0..69)]) # Peter Luschny, Aug 03 2015

G.f.: Product_{k>0} (1 - x^(2*k-1)) = Product_{k>0} 1 / (1 + x^k) = 1 + Sum_{k>0} (-x)^k / (Product_{i=1..k} (1 - x^i)).
a(n) = A027187(n)-A027193(n). [Fine]
This is the convolution inverse of A000009 (partitions into distinct parts) - i.e. the negation of the INVERTi transform of A000009. - Franklin T. Adams-Watters, Jan 06 2006
Expansion of chi(-x) = chi(-x^2) / chi(x) = phi(-x) / f(-x) = phi(-x^2) / f(x) = psi(-x) / f(-x^4) = f(-x) / f(-x^2) = f(-x^2) / psi(x) in powers of x where chi(), psi(), phi(), f() are Ramanujan theta functions.
Expansion of chi(x) * chi(-x) = f(x) / psi(x) = f(-x) / psi(-x) in powers of x^2 where chi(), psi(), f() are Ramanujan theta functions.
Expansion of f(-x, -x^5) / psi(x^3) = phi(-x^3) / f(x, x^2) in powers of x where phi(), psi(), f() are Ramanujan theta functions. - Michael Somos, Jun 03 2015
Given g.f. A(x), then B(q) = A(q^3)^8 / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u^2*v + 16*u - v^2.
G.f. A(x) satisfies A(x^2) = A(x) * A(-x).
G.f. is a period 1 Fourier series which satisfies f(-1 / (1152 t)) = 2^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A000009.
Euler transform of period 2 sequence [ -1, 0, ...].
Expansion of q^(1/24) * f1(t) in powers of q = exp(Pi i t) where f1() is a Weber function.
a(n) = (-1)^n * A000700(n). a(2*n) = A069910(n). a(2*n + 1) = - A069911(n).
a(n) ~ (-1)^n * exp(Pi*sqrt(n/6)) / (2*24^(1/4)*n^(3/4)). - Vaclav Kotesovec, Feb 25 2015
a(n) = Sum_{k = 0..n} (-1)^k*A072233(n,k). - Peter Luschny, Aug 03 2015
G.f.: Sum_{k>=0} (-1)^k * x^k^2 / (Product_{i=1..k} (1 - x^(2*i))). - Michael Somos, Aug 20 2015
Sum_{j>=0} a(n-A000217(j)) = 0 if n not in A152749, = (-1)^k if n is in A152749, n=k*(3*k-1). [Merca, Corollary 4.1] - R. J. Mathar, Jun 18 2016
a(n) = -(1/n)*Sum_{k=1..n} A000593(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017
G.f.: exp(Sum_{k>=1} (-1)^k*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, May 28 2018
G.f.: (1 - x) * Sum_{k >= 0} (-1)^k * x^(k*(k+2)) / (Product_{i = 1..k} (1 - x^(2*i))) = (1 - x)*(1 - x^3) * Sum_{k >= 0} (-1)^k * x^(k*(k+4)) / (Product_{i = 1..k} (1 - x^(2*i)))= (1 - x)*(1 - x^3)*(1 - x^5) *
Sum_{k >= 0} (-1)^k * x^(k*(k+6)) / (Product_{i = 1..k} (1 - x^(2*i))) = .... - Peter Bala, Jan 15 2021

A005875 Theta series of simple cubic lattice; also number of ways of writing a nonnegative integer n as a sum of 3 squares (zero being allowed).

Original entry on oeis.org

1, 6, 12, 8, 6, 24, 24, 0, 12, 30, 24, 24, 8, 24, 48, 0, 6, 48, 36, 24, 24, 48, 24, 0, 24, 30, 72, 32, 0, 72, 48, 0, 12, 48, 48, 48, 30, 24, 72, 0, 24, 96, 48, 24, 24, 72, 48, 0, 8, 54, 84, 48, 24, 72, 96, 0, 48, 48, 24, 72, 0, 72, 96, 0, 6, 96, 96, 24, 48, 96, 48, 0, 36, 48, 120

Offset: 0

N. J. A. Sloane

Number of ordered triples (i, j, k) of integers such that n = i^2 + j^2 + k^2.
The Madelung Coulomb energy for alternating unit charges in the simple cubic lattice is Sum_{n>=1} (-1)^n*a(n)/sqrt(n) = -A085469. - R. J. Mathar, Apr 29 2006
a(A004215(k))=0 for k=1,2,3,... but no other elements of {a(n)} are zero. - Graeme McRae, Jan 15 2007

			Order and signs are taken into account: a(1) = 6 from 1 = (+-1)^2 + 0^2 + 0^2, a(2) = 12 from 2 = (+-1)^2 + (+-1)^2 + 0^2; a(3) = 8 from 3 = (+-1)^2 + (+-1)^2 + (+-1)^2, etc.
G.f. =  1 + 6*q + 12*q^2 + 8*q^3 + 6*q^4 + 24*q^5 + 24*q^6 + 12*q^8 + 30*q^9 + 24*q^10 + ...

H. Cohen, Number Theory, Vol. 1: Tools and Diophantine Equations, Springer-Verlag, 2007, p. 317.
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 107.
H. Davenport, The Higher Arithmetic. Cambridge Univ. Press, 7th ed., 1999, Chapter V.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 3, p. 109.
E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 54.
L. Kronecker, Crelle, Vol. LVII (1860), p. 248; Werke, Vol. IV, p. 188.
C. J. Moreno and S. S. Wagstaff, Jr., Sums of Squares of Integers, Chapman and Hall, 2006, p. 43.
T. Nagell, Introduction to Number Theory, Wiley, 1951, p. 194.
W. Sierpiński, 1925. Teorja Liczb. pp. 1-410 (p.61).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
H. J. S. Smith, Report on the Theory of Numbers, reprinted in Vol. 1 of his Collected Math. Papers, Chelsea, NY, 1979, see p. 338, Eq. (B').

T. D. Noe, Table of n, a(n) for n = 0..10000
George E. Andrews, The Bhargava-Adiga Summation and Partitions, 2016. See Lemma 2.1.
P. T. Bateman, On the representations of a number as the sum of three squares, Trans. Amer. Math. Soc. 71 (1951), 70-101.
S. Bhargava and C. Adiga, A basic bilateral series summation formula and its applications, Integral Transforms and Special Functions, 2 (1994), 165-184.
J. M. Borwein, K-K S. Choi, On Dirichlet series for sums of squares, Raman. J. 7 (2003) 95-127
S. K. K. Choi, A. V. Kumchev, and R. Osburn, On sums of three squares, arXiv:math/0502007 [math.NT], 2005.
M. Doring, J. Haidenbauer, U.-G. Meissner, and A. Rusetsky, Dynamical coupled-channel approaches on a momentum lattice, arXiv:1108.0676 [hep-lat], 2011.
J. A. Ewell, Recursive determination of the enumerator for sums of three squares, Internat. J. Math. and Math. Sci, 24 (2000), 529-532.
O. Fraser and B. Gordon, On representing a square as the sum of three squares, Amer. Math. Monthly, 76 (1969), 922-923.
M. D. Hirschhorn and J. A. Sellers, On Representations of a Number as a Sum of Three Squares, Discrete Mathematics 199 (1999), 85-101.
M. D. Hirschhorn and J. A. Sellers, On Representations Of A Number As A Sum Of Three Squares
A. Martinez Torres, L. R. Dai, C. Koren, D. Jido, and E. Oset, The KD, eta D_s interaction in finite volume and the D_{s^*0}(2317) resonance, arXiv:1109.0396 [hep-lat], 2011.
R. J. Mathar, Hierarchical Subdivision of the Simple Cubic Lattice, arXiv:1309.3705 [math.MG], 2013.
S. C. Milne, Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions and Schur functions, Ramanujan J., 6 (2002), 7-149.
J. L. Mordell, The Representation Of Integers By Three Positive Squares, Michigan Math. J. 7(3): 289-290 (1960).
Eric T. Mortenson, A Kronecker-type identity and the representations of a number as a sum of three squares, arXiv:1702.01627 [math.NT], 2017.
G. Nebe and N. J. A. Sloane, Home page for this lattice
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Theta Series
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for sequences related to sums of squares

Row d=3 of A122141 and of A319574, 3rd column of A286815.
Cf. A074590 (primitive solutions), A117609 (partial sums), A004215 (positions of zeros).
Analog for 4 squares: A000118.
x^2+y^2+k*z^2: A005875, A014455, A034933, A169783, A169784.
Cf. A000164, A004013, A004015, A008443, A045828, A045831, A045834, A213384.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

# JacobiTheta3 is defined in A000122.
A005875List(len) = JacobiTheta3(len, 3)
A005875List(75) |> println # Peter Luschny, Mar 12 2018

Basis( ModularForms( Gamma1(4), 3/2), 75) [1]; /* Michael Somos, Jun 25 2014 */

(sum(x^(m^2),m=-10..10))^3; seq(coeff(%,x,n), n=0..50);
Alternative:
A005875list := proc(len) series(JacobiTheta3(0, x)^3, x, len+1);
seq(coeff(%, x, j), j=0..len-1) end: A005875list(75); # Peter Luschny, Oct 02 2018

SquaresR[3,Range[0,80]] (* Harvey P. Dale, Jul 21 2011 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^3, {q, 0, n}]; (* Michael Somos, Jun 25 2014 *)
a[ n_] := Length @ FindInstance[ n == x^2 + y^2 + z^2, {x, y, z}, Integers, 10^9]; (* Michael Somos, May 21 2015 *)
QP = QPochhammer; CoefficientList[(QP[q^2]^5/(QP[q]*QP[q^4])^2)^3 + O[q]^80, q] (* Jean-François Alcover, Nov 24 2015 *)

{a(n) = if( n<0, 0, polcoeff( sum( k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n))^3, n))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^5 / (eta(x + A) * eta(x^4 + A))^2)^3, n))}; /* Michael Somos, Jun 03 2012 */

{a(n) = my(G); if( n<0, 0, G = [ 1, 0, 0; 0, 1, 0; 0, 0, 1]; polcoeff( 1 + 2 * x * Ser( qfrep( G, n)), n))}; /* Michael Somos, May 21 2015 */

# uses Python code for A004018
from math import isqrt
def A005875(n): return A004018(n)+(sum(A004018(n-k**2) for k in range(1,isqrt(n)+1))<<1) # Chai Wah Wu, Jun 21 2024

Q = DiagonalQuadraticForm(ZZ, [1]*3)
Q.representation_number_list(75) # Peter Luschny, Jun 20 2014

A number n is representable as the sum of 3 squares iff n is not of the form 4^a (8k+7) (cf. A000378).
There is a classical formula (essentially due to Gauss):
For sums of 3 squares r_3(n): write (uniquely) -n=D(2^vf)^2, with D<0 fundamental discriminant, f odd, v>=-1. Then r_3(n) = 12L((D/.),0)(1-(D/2)) Sum_{d | f} mu(d)(D/d)sigma(f/d).
Here mu is the Moebius function, (D/2) and (D/d) are Kronecker-Legendre symbols, sigma is the sum of divisors function, L((D/.),0)=h(D)/(w(D)/2) is the value at 0 of the L function of the quadratic character (D/.), equal to the class number h(D) divided by 2 or 3 in the special cases D=-4 and -3. - Henri Cohen (Henri.Cohen(AT)math.u-bordeaux1.fr), May 12 2010
a(n) = 3*T(n) if n == 1,2,5,6 mod 8, = 2*T(n) if n == 3 mod 8, = 0 if n == 7 mod 8 and = a(n/4) if n == 0 mod 4, where T(n) = A117726(n). [Moreno-Wagstaff].
"If 12E(n) is the number of representations of n as a sum of three squares, then E(n) = 2F(n) - G(n) where G(n) = number of classes of determinant -n, F(n) = number of uneven classes." - Dickson, quoting Kronecker. [Cf. A117726.]
a(n) = Sum_{d^2|n} b(n/d^2), where b() = A074590() gives the number of primitive solutions.
Expansion of phi(q)^3 in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Oct 25 2006.
Euler transform of period 4 sequence [ 6, -9, 6, -3, ...]. - Michael Somos, Oct 25 2006
G.f.: (Sum_{k in Z} x^(k^2))^3.
a(8*n + 7) = 0. a(4*n) = a(n).
a(n) = A004015(2*n) = A014455(2*n) = A004013(4*n) = A169783(4*n). a(4*n + 1) = 6 * A045834(n). a(8*n + 3) = 8 * A008443(n). a(8*n + 5) = 24 * A045831(n). - Michael Somos, Jun 03 2012
a(4*n + 2) = 12 * A045828(n). - Michael Somos, Sep 03 2014
a(n) = (-1)^n * A213384(n). - Michael Somos, May 21 2015
a(n) = (6/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - Seiichi Manyama, May 27 2017
a(n) = A004018(n) + 2*Sum_{k=1..floor(sqrt(n))} A004018(n - k^2). - Daniel Suteu, Aug 27 2021
Convolution cube of A000122. Convolution of A004018 and A000122. - R. J. Mathar, Aug 03 2025

More terms from James Sellers, Aug 22 2000

A058696 Number of ways to partition 2n into positive integers.

Original entry on oeis.org

1, 2, 5, 11, 22, 42, 77, 135, 231, 385, 627, 1002, 1575, 2436, 3718, 5604, 8349, 12310, 17977, 26015, 37338, 53174, 75175, 105558, 147273, 204226, 281589, 386155, 526823, 715220, 966467, 1300156, 1741630, 2323520, 3087735, 4087968, 5392783, 7089500, 9289091

Offset: 0

N. J. A. Sloane, Dec 31 2000

nonn

A bisection of A000041, the other one is A058695.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). - Michael Somos, Feb 16 2014
a(n) is the number of partitions of 3n-2 having n as a part, for n >= 1.  Also, a(n+1) is the number of partitions of 3n having n as a part, for n >= 1.  - Clark Kimberling, Mar 02 2014

			G.f. = 1 + 2*x + 5*x^2 + 11*x^3 + 22*x^4 + 42*x^5 + 77*x^6 + 135*x^7 + ...
G.f. = q^-1 + 2*q^47 + 5*q^95 + 11*q^143 + 22*q^191 + 42*q^239 + 77*q^287 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Roland Bacher and Pierre De La Harpe, Conjugacy growth series of some infinitely generated groups, International Mathematics Research Notices, 2016, pp.1-53. (hal-01285685v2)
K. Blum, Bounds on the Number of Graphical Partitions, arXiv:2103.03196 [math.CO], 2021. See Table on p. 7.
Álvaro Gutiérrez and Mercedes H. Rosas, Partial symmetries of iterated plethysms, arXiv:2201.00240 [math.CO], 2022.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000041, A035294, A058695.

a:= n-> combinat[numbpart](2*n):
seq(a(n), n=0..42);  # Alois P. Heinz, Jan 29 2020

nn=100;Table[CoefficientList[Series[Product[1/(1-x^i),{i,1,nn}],{x,0,nn}],x][[2i-1]],{i,1,nn/2}] (* Geoffrey Critzer, Sep 28 2013 *)
(* also *)
Table[PartitionsP[2 n], {n, 0, 40}] (* Clark Kimberling, Mar 02 2014 *)
(* also *)
Table[Count[IntegerPartitions[3 n - 2], p_ /; MemberQ[p, n]], {n, 20}]   (* Clark Kimberling, Mar 02 2014 *)
nmax = 60; CoefficientList[Series[Product[(1 + x^(8*k-4))*(1 + x^(8*k))*(1 + x^k)^2/((1 + x^(8*k-1))*(1 + x^(8*k-7))*(1 - x^k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 17 2016 *)

combinat::partitions::count(2*i) $i=0..54 // Zerinvary Lajos, Apr 16 2007

{a(n) = if( n<0, 0, polcoeff( 1 / eta(x + O(x^(2*n + 1))), 2*n))}; /* Michael Somos, Apr 25 2003 */

a(n) = numbpart(2*n); \\ Michel Marcus, Sep 28 2013

Expansion of f(x^3, x^5) / f(-x)^2 in powers of x where f() is a Ramanujan theta function. - Michael Somos, Feb 16 2014
Euler transform of period 16 sequence [ 2, 2, 3, 2, 3, 1, 2, 1, 2, 1, 3, 2, 3, 2, 2, 1, ...]. - Michael Somos,  Apr 25 2003
a(n) = A000041(2*n).
Convolution of A000041 and A035294. - Michael Somos, Feb 16 2014
G.f.: Product_{k>=1} (1 + x^(8*k-4)) * (1 + x^(8*k)) * (1 + x^k)^2 / ((1 + x^(8*k-1)) * (1 + x^(8*k-7)) * (1 - x^k)). - Vaclav Kotesovec, Nov 17 2016
a(n) ~ exp(2*Pi*sqrt(n/3)) / (8*sqrt(3)*n). - Vaclav Kotesovec, Feb 16 2022

A067659 Number of partitions of n into distinct parts such that number of parts is odd.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 11, 14, 16, 19, 23, 27, 32, 38, 44, 52, 61, 71, 82, 96, 111, 128, 148, 170, 195, 224, 256, 293, 334, 380, 432, 491, 557, 630, 713, 805, 908, 1024, 1152, 1295, 1455, 1632, 1829, 2048, 2291, 2560, 2859, 3189, 3554, 3958, 4404

Offset: 0

Naohiro Nomoto, Feb 23 2002

Ramanujan theta functions: phi(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

			From _Gus Wiseman_, Jan 09 2021: (Start)
The a(5) = 1 through a(15) = 14 partitions (A-F = 10..15):
  5   6     7     8     9     A     B     C     D     E     F
      321   421   431   432   532   542   543   643   653   654
                  521   531   541   632   642   652   743   753
                        621   631   641   651   742   752   762
                              721   731   732   751   761   843
                                    821   741   832   842   852
                                          831   841   851   861
                                          921   931   932   942
                                                A21   941   951
                                                      A31   A32
                                                      B21   A41
                                                            B31
                                                            C21
                                                            54321
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Joerg Arndt, Matters Computational (The Fxtbook), end of section 16.4.2 "Partitions into distinct parts", pp.348ff
Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160 (March 2016), Pages 60-75, function q_o(n).
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Dominates A000009.
Numbers with these strict partitions as binary indices are A000069.
The non-strict version is A027193.
The Heinz numbers of these partitions are A030059.
The even version is A067661.
The version for rank is A117193, with non-strict version A101707.
The ordered version is A332304, with non-strict version A166444.
Other cases of odd length:
- A024429 counts set partitions of odd length.
- A089677 counts ordered set partitions of odd length.
- A174726 counts ordered factorizations of odd length.
- A339890 counts factorizations of odd length.
A008289 counts strict partitions by sum and length.
A026804 counts partitions whose least part is odd, with strict case A026832.
Cf. A000700, A027187, A030229, A117192, A332305.

b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
      `if`(n=0, t, add(b(n-i*j, i-1, abs(t-j)), j=0..min(n/i, 1))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..80);  # Alois P. Heinz, Apr 01 2014

b[n_, i_, t_] := b[n, i, t] = If[n > i*(i + 1)/2, 0, If[n == 0, t, Sum[b[n - i*j, i - 1, Abs[t - j]], {j, 0, Min[n/i, 1]}]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Jan 16 2015, after Alois P. Heinz *)
CoefficientList[Normal[Series[(QPochhammer[-x, x]-QPochhammer[x])/2, {x, 0, 100}]], x] (* Andrey Zabolotskiy, Apr 12 2017 *)
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&OddQ[Length[#]]&]],{n,0,30}] (* Gus Wiseman, Jan 09 2021 *)

{a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( (eta(x^2+A)/eta(x+A) - eta(x+A))/2, n))} /* Michael Somos, Feb 14 2006 */

N=66;  q='q+O('q^N);  S=1+2*sqrtint(N);
gf=sum(n=1,S, (n%2!=0) * q^(n*(n+1)/2) / prod(k=1,n, 1-q^k ) );
concat( [0], Vec(gf) )  /* Joerg Arndt, Oct 20 2012 */

N=66;  q='q+O('q^N);  S=1+sqrtint(N);
gf=sum(n=1, S, q^(2*n^2-n) / prod(k=1, 2*n-1, 1-q^k ) );
concat( [0], Vec(gf) )  \\ Joerg Arndt, Apr 01 2014

For g.f. see under A067661.
a(n) = (A000009(n)-A010815(n))/2. - Vladeta Jovovic, Feb 24 2002
Expansion of (1-phi(-q))/(2*chi(-q)) in powers of q where phi(),chi() are Ramanujan theta functions. - Michael Somos, Feb 14 2006
G.f.: sum(n>=1, q^(2*n^2-n) / prod(k=1..2*n-1, 1-q^k ) ). [Joerg Arndt, Apr 01 2014]
a(n) = A067661(n) - A010815(n). - Andrey Zabolotskiy, Apr 12 2017
A000009(n) = a(n) + A067661(n). - Gus Wiseman, Jan 09 2021

A002448 Expansion of Jacobi theta function theta_4(x).

Original entry on oeis.org

1, -2, 0, 0, 2, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

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

Number 2 of the 14 primitive eta-products which are holomorphic modular forms of weight 1/2 listed by D. Zagier on page 30 of "The 1-2-3 of Modular Forms". - Michael Somos, May 04 2016

			G.f. = 1 - 2*q + 2*q^4 - 2*q^9 + 2*q^16 - 2*q^25 + 2*q^36 - 2*q^49 + ...

Richard Bellman, A Brief Introduction to Theta Functions, Dover, 2013.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 93, Eq. (34.11), p. 6, Eq. (7.324).
J. Tannery and J. Molk, Eléments de la Théorie des Fonctions Elliptiques, Vol. 2, Gauthier-Villars, Paris, 1902; Chelsea, NY, 1972, see p. 27.
E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge Univ. Press, 4th ed., 1963, p. 464.

T. D. Noe, Table of n, a(n) for n = 0..10000
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, p. 103.
J. W. L. Glaisher, On the deduction of series from infinite products, Messenger of Math., 2 (1873), p. 141.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Jacobi Theta Functions
Eric Weisstein's World of Mathematics, q-Series Identities
D. Zagier, Elliptic modular forms and their applications in "The 1-2-3 of modular forms", Springer-Verlag, 2008

Cf. A000122, A000203, A010054, A089802.

using Nemo
function JacobiTheta4(len, r)
    R, x = PolynomialRing(ZZ, "x")
    e = theta_qexp(r, len, -x)
    [fmpz(coeff(e, j)) for j in 0:len - 1] end
A002448List(len) = JacobiTheta4(len, 1)
A002448List(105) |> println # Peter Luschny, Mar 12 2018

Maple
```
Sum((-x)^(m^2),m=-10..10);
```

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q], {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
QP = QPochhammer; s = QP[q]^2/QP[q^2] + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Dec 01 2015, adapted from PARI *)

{a(n) = if( n<0, 0, (-1)^n * issquare(n) * 2 - (n==0))}; /* Michael Somos, Jun 17 1999 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 / eta(x^2 + A), n))}; /* Michael Somos, May 01 2003 */

from sympy.ntheory.primetest import is_square
def A002448(n): return (-is_square(n) if n&1 else is_square(n))<<1 if n else 1 # Chai Wah Wu, May 17 2023

Expansion of phi(-q) in powers of q where phi() is a Ramanujan theta function.
Expansion of eta(q)^2 / eta(q^2) in powers of q. - Michael Somos, May 01 2003
Expansion of 2 * sqrt( k' * K / (2 Pi) ) in powers of q. - Michael Somos, Nov 30 2013
Euler transform of period 2 sequence [ -2, -1, ...]. - Michael Somos, May 01 2003
G.f.: Sum_{k in Z} (-1)^k * x^(k^2) = Product_{k>0} (1 - x^k) / (1 + x^k). - Michael Somos, May 01 2003.
G.f.: 1 - 2 Sum_{k>0} x^k/(1 - x^k) Product_{j=1..k} (1 - x^j) / (1 + x^j). - Michael Somos, Apr 12 2012
a(n) = -2 * b(n) where b(n) is multiplicative and b(2^e) = (-1)^(e/2) if e even, b(p^e) = 1 if p>2 and e even, otherwise 0. - Michael Somos, Jul 07 2006
a(3*n + 1) = -2 * A089802(n), a(9*n) = a(n). - Michael Somos, Jul 07 2006
a(3*n + 2) = a(4*n + 2) = a(4*n + 3) = 0. a(4*n) = A000122(n). a(n) = (-1)^n * A000122(n). a(8*n + 1) = -2 * A010054(n). - Michael Somos, Apr 12 2012
For n > 0, a(n) = 2*(floor(sqrt(n))-floor(sqrt(n-1)))*(-1)^(floor(sqrt(n))). - Mikael Aaltonen, Jan 17 2015
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 32^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A010054. - Michael Somos, May 05 2016
a(0) = 1, a(n) = -(2/n)*Sum_{k=1..n} A002131(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 29 2017
G.f.: exp(Sum_{k>=1} (sigma(k) - sigma(2*k))*x^k/k). - Ilya Gutkovskiy, Sep 19 2018
From Peter Bala, Feb 19 2021: (Start)
G.f: A(q) = eta(q^2)^5 / ( eta(-q)*eta(q^4) )^2.
A(q) = 1 + 2*Sum_{n >= 1} (-1)^n*q^(n*(n+1)/2)/( (1 + q^n) * Product_{k = 1..n} 1 - q^k ).
A(-q)^2 = 1 + 4*Sum_{n >= 1} (-1)^(n+1)*q^(2*n-1)/(1 - q^(2*n-1)), which gives the number of representations of an integer as a sum of two squares. See, for example, Fine, 26.63.
The unsigned sequence has the g.f. 1 + 2*Sum_{n >= 1} q^(n*(n+1)/2) * ( Product_{k = 1..n-1} 1 + q^k ) /( Product_{k = 1..n} 1 + q^(2*k) ) = 1 + 2*q + 2*q^4 + 2*q^9 + .... See Fine, equation 14.43. (End)
Form Peter Bala, Sep 27 2023: (Start)
G.f. A(q) satisfies A(q)*A(-q) = A(q^2)^2.
A(q) = Sum_{n >= 1} (-q)^(n-1)*Product_{k >= n} 1 - q^k. (End)

A033845 Numbers k of the form 2^i*3^j, where i and j >= 1.

Original entry on oeis.org

6, 12, 18, 24, 36, 48, 54, 72, 96, 108, 144, 162, 192, 216, 288, 324, 384, 432, 486, 576, 648, 768, 864, 972, 1152, 1296, 1458, 1536, 1728, 1944, 2304, 2592, 2916, 3072, 3456, 3888, 4374, 4608, 5184, 5832, 6144, 6912, 7776, 8748, 9216, 10368, 11664

Offset: 1

Jeff Burch

This sequence is easily confused with A003586, which gives numbers of the form 2^i*3^j with i, j >= 0, and is one-sixth of the present sequence. . Don't simply say "numbers of the form 2^i*3^j", but specify which sequence you mean. - N. J. A. Sloane, May 26 2024
Solutions to phi(n)=n/3 [See J-M. de Koninck & A. Mercier, problème 733].
Numbers n such that Sum_{d prime divisor of n} 1/d = 5/6. - Benoit Cloitre, Apr 13 2002
Also n such that Sum_{d|n} mu(d)^2/d = 2. - Benoit Cloitre, Apr 15 2002
Complement of A006899 with respect to A003586. - Reinhard Zumkeller, Sep 25 2008
In the sieve of Eratosthenes, if one crosses numbers off multiple times, these numbers are crossed off twice, first for 2 and then for 3. - Alonso del Arte, Aug 22 2011
Subsequence of A051037. - Reinhard Zumkeller, Sep 13 2011
Numbers n such that Sum_{d|n} A008683(d)*A000041(d) = 7. - Carl Najafi, Oct 19 2011
Numbers n such that Sum_{d|n} A008683(d)*A000700(d) = 2. - Carl Najafi, Oct 20 2011
Solutions to the equation A001615(x) = 2x. - Enrique Pérez Herrero, Jan 02 2012
So these numbers are called Psi-perfect numbers [see J-M. de Koninck & A. Mercier, problème 654]. - Bernard Schott, Nov 20 2020

J-M. de Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Ellipses, 2004, Problème 733, page 94.
J-M. de Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Ellipses, 2004, Problème 654, page 85.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Subsequence of A000423, A003586, A051037, A256617.
Cf. A001615, A006899, A086410, A086411, A008683, A143201.
Cf. A191475, A191476.

import Data.Set (singleton, deleteFindMin, insert)
a033845 n = a033845_list !! (n-1)
a033845_list = f (singleton (2*3)) where
   f s = m : f (insert (2*m) $ insert (3*m) s') where
     (m,s') = deleteFindMin s
-- Reinhard Zumkeller, Sep 13 2011

mx = 12000; Sort@ Flatten@ Table[2^i*3^j, {i, Log[2, mx]}, {j, Log[3, mx/2^i]}] (* Robert G. Wilson v, Aug 17 2012 *)

list(lim)=my(v=List(), N); for(n=0, log(lim\2)\log(3), N=6*3^n; while(N<=lim, listput(v, N); N<<=1)); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jan 02 2012

from sympy import integer_log
def A033845(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum((x//3**i).bit_length() for i in range(integer_log(x,3)[0]+1))
    return 6*bisection(f,n,n) # Chai Wah Wu, Sep 15 2024

# faster for initial segment of sequence
import heapq
from itertools import islice
def A033845gen(): # generator of terms
    v, oldv, h, psmooth_primes, = 1, 0, [1], [2, 3]
    while True:
        v = heapq.heappop(h)
        if v != oldv:
            yield 6*v
            oldv = v
            for p in psmooth_primes:
                heapq.heappush(h, v*p)
print(list(islice(A033845gen(), 50))) # Michael S. Branicky, Sep 18 2024

Six times the 3-smooth numbers (A003586). - Ralf Stephan, Apr 16 2004
A086411(a(n)) - A086410(a(n)) = 1. - Reinhard Zumkeller, Sep 25 2008
A143201(a(n)) = 2. - Reinhard Zumkeller, Sep 13 2011
a(n) = 2^A191475(n) * 3^A191476(n). - Zak Seidov, Nov 01 2013
Sum_{n>=1} 1/a(n) = 1/2. - Amiram Eldar, Oct 13 2020

Edited by N. J. A. Sloane, Jan 31 2010 and May 26 2024.

A339890 Number of odd-length factorizations of n into factors > 1.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Dec 28 2020

nonn

			The a(n) factorizations for n = 24, 48, 60, 72, 96, 120:
  24      48          60       72          96          120
  2*2*6   2*3*8       2*5*6    2*4*9       2*6*8       3*5*8
  2*3*4   2*4*6       3*4*5    2*6*6       3*4*8       4*5*6
          3*4*4       2*2*15   3*3*8       4*4*6       2*2*30
          2*2*12      2*3*10   3*4*6       2*2*24      2*3*20
          2*2*2*2*3            2*2*18      2*3*16      2*4*15
                               2*3*12      2*4*12      2*5*12
                               2*2*2*3*3   2*2*2*2*6   2*6*10
                                           2*2*2*3*4   3*4*10
                                                       2*2*2*3*5

Alois P. Heinz, Table of n, a(n) for n = 1..20000

The case of set partitions (or n squarefree) is A024429.
The case of partitions (or prime powers) is A027193.
The ordered version is A174726 (even: A174725).
The remaining (even-length) factorizations are counted by A339846.
A000009 counts partitions into odd parts, ranked by A066208.
A001055 counts factorizations, with strict case A045778.
A027193 counts partitions of odd length, ranked by A026424.
A058695 counts partitions of odd numbers, ranked by A300063.
A160786 counts odd-length partitions of odd numbers, ranked by A300272.
A316439 counts factorizations by product and length.
A340101 counts factorizations into odd factors.
A340102 counts odd-length factorizations into odd factors.
Cf. A000700, A002033, A027187, A028260, A074206, A078408, A236914, A320732.

g:= proc(n, k, t) option remember; `if`(n>k, 0, t)+
      `if`(isprime(n), 0, add(`if`(d>k, 0, g(n/d, d, 1-t)),
          d=numtheory[divisors](n) minus {1, n}))
    end:
a:= n-> `if`(n<2, 0, g(n$2, 1)):
seq(a(n), n=1..100);  # Alois P. Heinz, Dec 30 2020

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],OddQ@Length[#]&]],{n,100}]

a(n) + A339846(n) = A001055(n).

A008438 Sum of divisors of 2*n + 1.

Original entry on oeis.org

1, 4, 6, 8, 13, 12, 14, 24, 18, 20, 32, 24, 31, 40, 30, 32, 48, 48, 38, 56, 42, 44, 78, 48, 57, 72, 54, 72, 80, 60, 62, 104, 84, 68, 96, 72, 74, 124, 96, 80, 121, 84, 108, 120, 90, 112, 128, 120, 98, 156, 102, 104, 192, 108, 110, 152, 114, 144, 182, 144, 133, 168

Offset: 0

N. J. A. Sloane

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number of ways of writing n as the sum of 4 triangular numbers.
Bisection of A000203. - Omar E. Pol, Mar 14 2012
a(n) is also the total number of parts in all partitions of 2*n + 1 into equal parts. - Omar E. Pol, Feb 14 2021

			Divisors of 9 are 1,3,9, so a(4)=1+3+9=13.
F_2(z) = eta(4z)^8/eta(2z)^4 = q + 4q^3 + 6q^5 +8q^7 + 13q^9 + ...
G.f. = 1 + 4*x + 6*x^2 + 8*x^3 + 13*x^4 + 12*x^5 + 14*x^6 + 24*x^7 + 18*x^8 + 20*x^9 + ...
B(q) = q + 4*q^3 + 6*q^5 + 8*q^7 + 13*q^9 + 12*q^11 + 14*q^13 + 24*q^15 + 18*q^17 + ...

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 139 Ex. (iii).
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 19 eq. (6), and p. 283 eq. (8).
W. Dunham, Euler: The Master of Us All, The Mathematical Association of America Inc., Washington, D.C., 1999, p. 12.
H. M. Farkas, I. Kra, Cosines and triangular numbers, Rev. Roumaine Math. Pures Appl., 46 (2001), 37-43.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 79, Eq. (32.31).
N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, 1984, see p. 184, Prop. 4, F(z).
G. Polya, Induction and Analogy in Mathematics, vol. 1 of Mathematics and Plausible Reasoning, Princeton Univ. Press, 1954, page 92 ff.

N. J. A. Sloane, Table of n, a(n) for n = 0..20000 (first 10000 terms from T. D. Noe)
H. Cohen, Sums involving the values at negative integers of L-functions of quadratic characters, Math. Ann. 217 (1975), no. 3, 271-285. MR0382192 (52 #3080)
M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211
Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. I wrote 2005 on the first page but the internal evidence suggests 1997.]
K. Ono, S. Robins and P. T. Wahl, On the representation of integers as sums of triangular numbers, Aequationes mathematicae, August 1995, Volume 50, Issue 1-2, pp 73-94. Theorem 3 [Legendre].
H. Rosengren, Sums of triangular numbers from the Frobenius determinant, arXiv:math/0504272 [math.NT], 2005.
Michael Somos, Introduction to Ramanujan theta functions
Min Wang and Zhi-Hong Sun, On the number of representations of n as a linear combination of four triangular numbers II, arXiv:1511.00478 [math.NT], 2015.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
K. S. Williams, The parents of Jacobi's four squares theorem are unique, Amer. Math. Monthly, 120 (2013), 329-345.

Cf. A000118, A000593, A005879, A096727, A115607, A129588, A225699/A225700.
Number of ways of writing n as a sum of k triangular numbers, for k=1,...: A010054, A008441, A008443, A008438, A008439, A008440, A226252, A007331, A226253, A226254, A226255, A014787, A014809.
Cf. A000203, A002131, A005408, A062731, A099774, A125061, A138741.

a008438 = a000203 . a005408  -- Reinhard Zumkeller, Sep 22 2014

Basis( ModularForms( Gamma0(4), 2), 124) [2]; /* Michael Somos, Jun 12 2014 */

[DivisorSigma(1, 2*n+1): n in [0..70]]; // Vincenzo Librandi, Aug 01 2017

A008438 := proc(n) numtheory[sigma](2*n+1) ; end proc: # R. J. Mathar, Mar 23 2011

DivisorSigma[1, 2 # + 1] & /@ Range[0, 61] (* Ant King, Dec 02 2010 *)
a[ n_] := SeriesCoefficient[ D[ Series[ Log[ QPochhammer[ -x] / QPochhammer[ x]], {x, 0, 2 n + 1}], x], {x, 0 , 2n}]; (* Michael Somos, Oct 15 2019 *)

PARI
```
{a(n) = if( n<0, 0, sigma( 2*n + 1))};
```

{a(n) = if( n<0, 0, n = 2*n; polcoeff( sum( k=1, (sqrtint( 4*n + 1) + 1)\2, x^(k^2 - k), x * O(x^n))^4, n))}; /* Michael Somos, Sep 17 2004 */

{a(n) = my(A); if( n<0, 0, n = 2*n; A = x * O(x^n); polcoeff( (eta(x^4 + A)^2 / eta(x^2 + A))^4, n))}; /* Michael Somos, Sep 17 2004 */

ModularForms( Gamma0(4), 2, prec=124).1;  # Michael Somos, Jun 12 2014

Expansion of q^(-1/2) * (eta(q^2)^2 / eta(q))^4 = psi(q)^4 in powers of q where psi() is a Ramanujan theta function. - Michael Somos, Apr 11 2004
Expansion of Jacobi theta_2(q)^4 / (16*q) in powers of q^2. - Michael Somos, Apr 11 2004
Euler transform of period 2 sequence [4, -4, 4, -4, ...]. - Michael Somos, Apr 11 2004
a(n) = b(2*n + 1) where b() is multiplicative and b(2^e) = 0^n, b(p^e) =(p^(e+1) - 1) / (p - 1) if p>2. - Michael Somos, Jul 07 2004
Given g.f. A(x), then B(q) = q * A(q^2) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = v^3 + 8*w*v^2 + 16*w^2*v - u^2*w - Michael Somos, Apr 08 2005
Given g.f. A(x), then B(q) = q * A(q^2) satisfies 0 = f(B(q), B(q^3), B(q^9)) where f(u, v, w) = v^4 - 30*u*v^2*w + 12*u*v*w*(u + 9*w) - u*w*(u^2 + 9*w*u + 81*w^2).
Given g.f. A(x), then B(q) = q * A(q^2) satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u2^3 + u1^2*u6 + 3*u2*u3^2 + 27*u6^3 - u1*u2*u3 - 3*u1*u3*u6 - 7*u2^2*u6 - 21*u2*u6^2. - Michael Somos, May 30 2005
G.f.: Sum_{k>=0} (2k + 1) * x^k / (1 - x^(2k + 1)).
G.f.: (Product_{k>0} (1 - x^k) * (1 + x^k)^2)^4. - Michael Somos, Apr 11 2004
G.f. Sum_{k>=0} a(k) * x^(2*k + 1) = x * (Product_{k>0} (1 - x^(4*k))^2 / (1 - x^(2*k)))^4 = x * (Sum_{k>0} x^(k^2 - k))^4 = Sum_{k>0} k * (x^k / (1 - x^k) - 3 * x^(2*k) / (1 - x^(2*k)) + 2 * x^(4*k) / (1 - x^(4*k))). - Michael Somos, Jul 07 2004
Number of solutions of 2*n + 1 = (x^2 + y^2 + z^2 + w^2) / 4 in positive odd integers. - Michael Somos, Apr 11 2004
8 * a(n) = A005879(n) = A000118(2*n + 1). 16 * a(n) = A129588(n). a(n) = A000593(2*n + 1) = A115607(2*n + 1).
a(n) = A000203(2*n+1). - Omar E. Pol, Mar 14 2012
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = (1/4) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A096727. Michael Somos, Jun 12 2014
a(0) = 1, a(n) = (4/n)*Sum_{k=1..n} A002129(k)*a(n-k) for n > 0. - Seiichi Manyama, May 06 2017
G.f.: exp(Sum_{k>=1} 4*(x^k/k)/(1 + x^k)). - Ilya Gutkovskiy, Jul 31 2017
From Peter Bala, Jan 10 2021: (Start)
a(n) = A002131(2*n+1).
G.f.: Sum_{n >= 0} x^n*(1 + x^(2*n+1))/(1 - x^(2*n+1))^2. (End)
Sum_{k=1..n} a(k) ~ Pi^2 * n^2 / 8. - Vaclav Kotesovec, Aug 07 2022
Convolution of A125061 and A138741. - Michael Somos, Mar 04 2023

Comments from Len Smiley, Enoch Haga

A030203 Expansion of q^(-1/6) * eta(q) * eta(q^3) in powers of q.

Original entry on oeis.org

1, -1, -1, -1, 1, 2, -1, 2, 0, 0, -1, -1, -1, -1, 0, 1, -1, -1, 2, 0, 1, 2, 1, -1, 0, -1, 2, -1, 0, -1, -1, 0, -1, -1, 0, -1, -2, 2, 2, 0, -1, 1, 0, 1, 0, -1, 2, 2, 1, 0, -2, 2, -1, 0, -1, -1, -1, 1, -1, 0, 0, -1, -1, -1, 0, 0, 2, -2, -1, 0, -1, 1, 2, 2, 0, 0

Offset: 0

N. J. A. Sloane

sign

Number 65 of the 74 eta-quotients listed in Table I of Martin (1996).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f. = 1 - x - x^2 - x^3 + x^4 + 2*x^5 - x^6 + 2*x^7 - x^10 - x^11 - x^12 - x^13 + ...
G.f. = q - q^7 - q^13 - q^19 + q^25 + 2*q^31 - q^37 + 2*q^43 - q^61 - q^67 - ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
S. R. Finch, Powers of Euler's q-Series, (arXiv:math.NT/0701251).
M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Basis( CuspForms( Gamma1(108), 1), 452)[1]; /* Michael Somos, Jan 31 2015 */

a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^3], {x, 0, n}]; (* Michael Somos, Jan 31 2015 *)

{a(n) = my(A, p, e); if( n<0, 0, n = 6*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p<5, 0, p%6==5, (1 + (-1)^e) / 2, (p-1) / znorder( Mod(2, p))%3, kronecker( e+1, 3), e+1)))}; /* Michael Somos, Jul 27 2006 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^3 + A), n))}; /* Michael Somos, Jul 27 2006 */

Expansion of f(-x) * f(-x^3) where f(-x) := f(-x, -x^2) is a Ramanujan theta function. - Michael Somos, Jul 27 2006
Expansion of q^(-1/6) * sqrt(b(q) * c(q)/3) in powers of q where b(), c() are cubic AGM theta functions. - Michael Somos, Nov 01 2006
Euler transform of period 3 sequence [ -1, -1, -2, ...]. - Michael Somos, Jul 27 2006
Given g.f. A(x), then B(q) = (q * A(q^6))^2 satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = v^3 - u^2*w - 4*u*w^2. - Michael Somos, Jul 27 2006
a(n) = b(6*n + 1) where b(n) is multiplicative and b(2^e) = b(3^e) = 0^e, b(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6), b(p^e) = e+1 if p = x^2 + 27*y^2, b(p^e) = [1, -1, 0] depending on e (mod 3) otherwise.
G.f. is a period 1 Fourier series which satisfies f(-1 / (108 t)) = 108^(1/2) (t/i) f(t) where q = exp(2 Pi i t). - Michael Somos, Jan 22 2012
G.f.: Product_{k>0} (1 - x^k) * (1 - x^(3*k)).

cp's OEIS Frontend

A004018 Theta series of square lattice (or number of ways of writing n as a sum of 2 squares). Often denoted by r(n) or r_2(n).

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A081362 Expansion of q^(1/24) * eta(q) / eta(q^2) in powers of q.

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A005875 Theta series of simple cubic lattice; also number of ways of writing a nonnegative integer n as a sum of 3 squares (zero being allowed).

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A058696 Number of ways to partition 2n into positive integers.

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A067659 Number of partitions of n into distinct parts such that number of parts is odd.

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