A000143 -id:A000143 - cp's OEIS frontend

A340915 Number of ways to write n as an ordered sum of 8 squares of positive integers.

1, 0, 0, 8, 0, 0, 28, 0, 8, 56, 0, 56, 70, 0, 168, 64, 28, 280, 84, 168, 280, 176, 420, 224, 345, 560, 392, 616, 420, 848, 924, 336, 1246, 1064, 868, 1464, 988, 1680, 1820, 1120, 1904, 2464, 1932, 1904, 2870, 2752, 2772, 2912, 2892, 4256, 3640, 3248, 4480, 5040, 4760, 3696, 6120

Offset: 8

Ilya Gutkovskiy, Jan 31 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 8..20000
Index entries for sequences related to sums of squares

Cf. A000143, A000290, A010052, A025432, A045850, A063691, A063725, A063730, A340481, A340905, A340906, A340946, A340947.

Column k=8 of A337165.

b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add((s->
      `if`(s>n, 0, b(n-s, t-1)))(j^2), j=1..isqrt(n))))
    end:
a:= n-> b(n, 8):
seq(a(n), n=8..64);  # Alois P. Heinz, Jan 31 2021

nmax = 64; CoefficientList[Series[(EllipticTheta[3, 0, x] - 1)^8/256, {x, 0, nmax}], x] // Drop[#, 8] &

G.f.: (theta_3(x) - 1)^8 / 256, where theta_3() is the Jacobi theta function.

A341397 Number of integer solutions to (x_1)^2 + (x_2)^2 + ... + (x_8)^2 <= n.

Original entry on oeis.org

1, 17, 129, 577, 1713, 3729, 6865, 12369, 21697, 33809, 47921, 69233, 101041, 136209, 174737, 231185, 306049, 384673, 469457, 579217, 722353, 876465, 1025649, 1220337, 1481521, 1733537, 1979713, 2306753, 2697537, 3087777, 3482913, 3959585, 4558737, 5155473

Offset: 0

Ilya Gutkovskiy, Feb 10 2021

nonn

Partial sums of A000143.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for sequences related to sums of squares

Cf. A000122, A000143, A001650, A046895, A055407, A055414, A057655, A117609, A122510, A175360, A175361, A302860, A341396, A341398, A341399.

b:= proc(n, k) option remember; `if`(n=0, 1, `if`(n<0 or k<1, 0,
      b(n, k-1)+2*add(b(n-j^2, k-1), j=1..isqrt(n))))
    end:
a:= proc(n) option remember; b(n, 8)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..33);  # Alois P. Heinz, Feb 10 2021

nmax = 33; CoefficientList[Series[EllipticTheta[3, 0, x]^8/(1 - x), {x, 0, nmax}], x]
Table[SquaresR[8, n], {n, 0, 33}] // Accumulate

from math import prod
from sympy import factorint
def A341397(n): return (sum((prod((p**(3*(e+1))-(1 if p&1 else 15))//(p**3-1) for p, e in factorint(m).items()) for m in range(1,n+1)))<<4)+1 # Chai Wah Wu, Jun 21 2024

G.f.: theta_3(x)^8 / (1 - x).

a(n^2) = A055414(n).

A276285 Number of ways of writing n as a sum of 13 squares.

Original entry on oeis.org

1, 26, 312, 2288, 11466, 41808, 116688, 265408, 535704, 1031914, 1899664, 3214224, 5043376, 7801744, 12066912, 17689152, 24443978, 34039200, 48210760, 64966096, 83323344, 109157152, 145532816, 185245632, 227110416, 284788010, 363737712

Offset: 0

Ilya Gutkovskiy, Aug 27 2016

More generally, the ordinary generating function for the number of ways of writing n as a sum of k squares is theta_3(0, q)^k = 1 + 2*k*q + 2*(k - 1)*k*q^2 + (4/3)*(k - 2)*(k - 1)*k*q^3 + (2/3)*((k - 3)*(k - 2)*(k - 1) + 3)*k*q^4 + (4/15) *(k - 1)*k*(k^3 - 9*k^2 + 26*k - 9)*q^5 + ..., where theta is the Jacobi theta functions.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Sum of Squares Function
Eric Weisstein's World of Mathematics, Jacobi Theta Functions
Index entries for sequences related to sums of squares

13th column of A286815. - Seiichi Manyama, May 27 2017
Row d=13 of A122141.
Cf. Number of ways of writing n as a sum of k squares: A004018 (k = 2), A005875 (k = 3), A000118 (k = 4), A000132 (k = 5), A000141 (k = 6), A008451 (k = 7), A000143 (k = 8), A008452 (k = 9), A000144 (k = 10), A008453 (k = 11), A000145 (k = 12), this sequence (k = 13), A000152 (k = 16).

Mathematica
```
Table[SquaresR[13, n], {n, 0, 26}]
```

G.f.: theta_3(0,q)^13, where theta_3(x,q) is the third Jacobi theta function.

a(n) = (26/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - Seiichi Manyama, May 27 2017

A340999 Number of partitions of n into 8 distinct nonzero squares.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 0, 1, 0, 0, 2, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 3, 0, 0, 1

Offset: 204

Ilya Gutkovskiy, Feb 02 2021

nonn

Index entries for sequences related to sums of squares

Cf. A000143, A000290, A010052, A025432, A025441, A025442, A025443, A025444, A045850, A224983, A340915, A340988, A340998, A341000, A341001.

Column k=8 of A341040.

A008430 Theta series of D_8 lattice.

Original entry on oeis.org

1, 112, 1136, 3136, 9328, 14112, 31808, 38528, 74864, 84784, 143136, 149184, 261184, 246176, 390784, 395136, 599152, 550368, 859952, 768320, 1175328, 1078784, 1513152, 1362816, 2096192, 1764112, 2496928, 2289280, 3208832, 2731680, 4007808, 3336704

Offset: 0

N. J. A. Sloane

			1 + 112*q^2 + 1136*q^4 + 3136*q^6 + 9328*q^8 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..5000 from G. C. Greubel)
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, p. 118.
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006.
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

Cf. A000143, A008427 (dual), A109773.

a[n_] := 16*DivisorSum[n, #^3*(8 - Mod[#, 2]) &]; a[0] = 1; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Dec 02 2015, adapted from PARI *)

{a(n)=if(n<1, n==0, 16*sumdiv(n, d, d^3*(8-d%2)))} /* Michael Somos, Nov 03 2006 */

{a(n)=if(n<0, 0, n*=2; polcoeff( sum(k=1, sqrtint(n), 2*x^k^2, 1+x*O(x^n))^8, n))} /* Michael Somos, Nov 03 2006 */

G.f.: (theta_3(q^(1/2))^8 + theta_4(q^(1/2))^8)/2.

a(n) = A000143(2n).

A209942 Expansion of (psi(-x) * phi(x)^4)^2 in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, 14, 81, 238, 322, 0, -429, -82, 0, -2162, -3038, 1134, 2401, -2482, 0, 6958, 3332, 0, 1442, 0, 6561, 4508, -9758, 0, -1918, -18802, 0, -9362, -24638, 19278, 14641, -14756, 0, 0, 6562, 0, -1148, 33998, 26082, 20398, 0, 0, 28083, -49042, 0, 64078, -30268, 0

Offset: 0

Michael Somos, Mar 16 2012

sign

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

Number 60 of the 74 eta-quotients listed in Table I of Martin (1996).

			G.f. = 1 + 14*x + 81*x^2 + 238*x^3 + 322*x^4 - 429*x^6 - 82*x^7 - 2162*x^9 + ...
G.f. = q + 14*q^5 + 81*q^9 + 238*q^13 + 322*q^17 - 429*q^25 - 82*q^29 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
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, 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

Cf. A000143, A134343, A258771.

a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2]^19 / (QPochhammer[ x] QPochhammer[ x^4])^7)^2, {x, 0, n}]; (* Michael Somos, Jun 09 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( eta(x^2 + A)^19 / (eta(x + A) * eta(x^4 + A) )^7 )^2, n))};

Expansion of q^(-1/4) * ( eta(q^2)^19 / (eta(q) * eta(q^4) )^7 )^2 in powers of q.
Euler transform of period 4 sequence [ 14, -24, 14, -10, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 32768 (t/i)^5 f(t) where q = exp(2 Pi i t).
a(n) = b(4*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 * p^(2*e) if p == 3 (mod 4), b(p^e) = b(p) * b(p^(e-1)) - p^4 * b(p^(e-2)) otherwise.
a(9*n + 5) = a(9*n + 8) = 0. a(9*n + 2) = 81 * a(n). Convolution of A000143 and A134343.
Convolution square of A258771. - Michael Somos, Jun 09 2015

A004409 Expansion of ( Sum_{n = -infinity..infinity} x^(n^2) )^(-8).

Original entry on oeis.org

1, -16, 144, -960, 5264, -25056, 106944, -418176, 1520784, -5201232, 16871648, -52252992, 155341248, -445226848, 1234726272, -3323392128, 8704504976, -22234655520, 55498917840, -135595345600, 324759439584

Offset: 0

N. J. A. Sloane

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A000122, A000143.

nmax = 30; CoefficientList[Series[Product[((1 + (-x)^k)/(1 - (-x)^k))^8, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 18 2015 *)

q='q+O('q^99); Vec(((eta(q)*eta(q^4))^2/eta(q^2)^5)^8) \\ Altug Alkan, Sep 20 2018

a(n) ~ (-1)^n * exp(2*Pi*sqrt(2*n)) / (64*2^(3/4)*n^(11/4)). - Vaclav Kotesovec, Aug 18 2015
From Ilya Gutkovskiy, Sep 20 2018: (Start)
G.f.: 1/theta_3(x)^8, where theta_3() is the Jacobi theta function.
G.f.: Product_{k>=1} 1/((1 - x^(2*k))*(1 + x^(2*k-1))^2)^8. (End)

A177155 G.f.: exp( Integral (theta_3(x)^8-1)/(16x) dx ), where theta_3(x) = 1 + Sum_{n>=1} 2*x^(n^2) is a Jacobi theta function.

Original entry on oeis.org

1, 1, 4, 13, 35, 87, 217, 539, 1291, 2999, 6880, 15595, 34738, 76202, 165282, 354655, 752546, 1580514, 3289337, 6787085, 13887937, 28195434, 56824772, 113729640, 226104615, 446665922, 877063515, 1712252521, 3324245063, 6419561961

Offset: 0

Paul D. Hanna, May 03 2010, May 08 2010

nonn

Compare to g.f. of partitions in which no parts are multiples of 4:

g.f. of A001935 = exp( Integral (theta_3(x)^4-1)/(8x) dx ).

			G.f.: A(x) = 1 + x + 4*x^2 + 13*x^3 + 35*x^4 + 87*x^5 +...
log(A(x)) = x + 7*x^2/2 + 28*x^3/3 + 71*x^4/4 + 126*x^5/5 +...+ A008457(n)*x^n/n +...

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A138503, A008457, A001935, A000143, A307462.

nmax = 40; Abs[CoefficientList[Series[Product[1/(1 - x^k)^((-1)^k*k^2), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Apr 10 2019 *)
nmax = 40; CoefficientList[Series[Product[(1 + x^(2*k - 1))^((2*k - 1)^2)/(1 - x^(2*k))^(4*k^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 10 2019 *)

{a(n)=polcoeff(exp(sum(m=1,n, sumdiv(m,d,(-1)^(m-d)*d^3)*x^m/m)+x*O(x^n)),n)}

{a(n)=local(theta3=1+sum(m=1,sqrtint(2*n+2),2*x^(m^2)+x*O(x^n)));polcoeff(exp(intformal((theta3^8-1)/(16*x))),n)}

G.f.: exp( Sum_{n>=1} A008457(n)*x^n/n ) where A008457(n) = Sum_{d|n} (-1)^(n-d)*d^3.

a(n) ~ exp(2*Pi*n^(3/4)/3 - Zeta(3)/Pi^2) / (4*n^(5/8)). - Vaclav Kotesovec, Apr 10 2019

A166947 Number of ways of writing n as the sum of 2^n squares.

Original entry on oeis.org

1, 4, 24, 448, 29152, 6448000, 4799359488, 12099984537600, 104875315518635520, 3178565207840143938560, 342288453932192597037125632, 132776310046929259464457969090560

Offset: 0

Paul D. Hanna, Oct 25 2009

nonn

			G.f.: A(x) = 1 + 4*x + 24*x^2 + 448*x^3 + 29152*x^4 + 6448000*x^5 +...
Let F(x) = theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2),
then A(x) = 1 + log(F(2*x)) + log(F(4*x))^2/2! + log(F(8*x))^3/3! + ...
Illustrate a(n) = [x^n] F(x)^(2^n) by forming a table of
coefficients in powers F(x)^(2^n), which begin:
F^(2^0): [(1), 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, ...];
F^(2^1): [1, (4), 4, 0, 4, 8, 0, 0, 4, 4, 8, 0, 0, 8, 0, 0, ...];
F^(2^2): [1, 8, (24), 32, 24, 48, 96, 64, 24, 104, 144, 96, ...];
F^(2^3): [1, 16, 112, (448), 1136, 2016, 3136, 5504, 9328, ...];
F^(2^4): [1, 32, 480, 4480, (29152), 140736, 525952, 1580800, ...];
F^(2^5): [1, 64, 1984, 39680, 575424, (6448000), 58115328, ...];
F^(2^6): [1, 128, 8064, 333312, 10166144, 244000512, (4799359488), ...];
F^(2^7): [1, 256, 32512, 2731008, 170688256, 8466189824, 347119309824, (12099984537600), ...]; ...
and noting that the coefficients along the diagonal (in parenthesis)
form the initial terms of this sequence.

Cf. A158112, A000122, A004018, A000118, A000143, A000152, A022085.

Cf. variant: A166953 (n as the sum of 3^n squares). [From Paul D. Hanna, Oct 26 2009]

{a(n)=local(THETA3=1+2*sum(k=1,sqrtint(n),x^(k^2))+x*O(x^n));polcoeff(THETA3^(2^n),n)}

{a(n)=local(THETA3=1+2*sum(k=1,sqrtint(n),x^(k^2))+x*O(x^n)); polcoeff(sum(k=0,n,log(subst(THETA3,x,2^k*x))^k/k!),n)}

a(n) equals the coefficient of x^n in the (2^n)-th power of Jacobi theta_3(x).

G.f.: A(x) = Sum_{n>=0} log( theta_3(2^n*x) )^n/n! where theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2).

A374493 Number of ways of writing n^2 as a sum of 8 squares.

Original entry on oeis.org

1, 16, 1136, 12112, 74864, 252016, 859952, 1887888, 4793456, 8830096, 17893136, 28366288, 56672048, 77264112, 134040048, 190776112, 306783344, 386279728, 626936816, 752843856, 1179182864, 1429131216, 2014006448, 2368768912, 3628646192, 3937752016, 5485751952

Offset: 0

Seiichi Manyama, Jul 12 2024

nonn

Column k=8 of A302996.

Cf. A000143.

SquaresR[8, Range[0,30]^2] (* Paolo Xausa, Aug 21 2025 *)

a000143(n) = if(n==0, 1, 16*sumdiv(n, d, (-1)^(n+d)*d^3));
a(n) = a000143(n^2);

a(n) = [x^(n^2)] (Sum_{j=-n..n} x^(j^2))^8.
a(n) = A000143(n^2).
a(n) is divisible by 16 for n > 0.

cp's OEIS Frontend

A340915 Number of ways to write n as an ordered sum of 8 squares of positive integers.

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A341397 Number of integer solutions to (x_1)^2 + (x_2)^2 + ... + (x_8)^2 <= n.

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A276285 Number of ways of writing n as a sum of 13 squares.

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A340999 Number of partitions of n into 8 distinct nonzero squares.

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A008430 Theta series of D_8 lattice.

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A209942 Expansion of (psi(-x) * phi(x)^4)^2 in powers of x where phi(), psi() are Ramanujan theta functions.

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A004409 Expansion of ( Sum_{n = -infinity..infinity} x^(n^2) )^(-8).

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A177155 G.f.: exp( Integral (theta_3(x)^8-1)/(16x) dx ), where theta_3(x) = 1 + Sum_{n>=1} 2*x^(n^2) is a Jacobi theta function.

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