A186690 -id:A186690 - cp's OEIS frontend

A008452 Number of ways of writing n as a sum of 9 squares.

1, 18, 144, 672, 2034, 4320, 7392, 12672, 22608, 34802, 44640, 60768, 93984, 125280, 141120, 182400, 262386, 317376, 343536, 421344, 557280, 665280, 703584, 800640, 1068384, 1256562, 1234080, 1421184, 1851264, 2034720, 2057280, 2338560

Offset: 0

N. J. A. Sloane

nonn

E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 121.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 314.
Lomadze, G.A.: On the representations of natural numbers by sums of nine squares. Acta. Arith. 68(3), 245-253 (1994). (Russian). See Equation (3.6).

T. D. Noe, Table of n, a(n) for n = 0..10000
Shi-Chao Chen, Congruences for rs(n), Journal of Number Theory, Volume 130, Issue 9, September 2010, Pages 2028-2032.
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.
M. Peters, Sums of nine squares, Acta Arith., 102 (2002), 131-135.
Index entries for sequences related to sums of squares

Row d=9 of A122141 and of A319574, 9th column of A286815.

Cf. A008431.

(sum(x^(m^2),m=-10..10))^9;
# Alternative
A008452list := proc(len) series(JacobiTheta3(0, x)^9, x, len+1);
seq(coeff(%, x, j), j=0..len-1) end: A008452list(32); # Peter Luschny, Oct 02 2018

Table[SquaresR[9, n], {n, 0, 32}] (* Ray Chandler, Nov 28 2006 *)

# uses Python code from A000143
from math import isqrt
def A008452(n): return A000143(n)+(sum(A000143(n-k**2) for k in range(1,isqrt(n)+1))<<1) # Chai Wah Wu, Jun 23 2024

Q = DiagonalQuadraticForm(ZZ, [1]*9)
Q.representation_number_list(37) # Peter Luschny, Jun 20 2014

G.f.: theta_3(0,q)^9, where theta_3 is the 3rd Jacobi theta function. - Ilya Gutkovskiy, Jan 13 2017

a(n) = (18/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - Seiichi Manyama, May 27 2017

Extended by Ray Chandler, Nov 28 2006

A000144 Number of ways of writing n as a sum of 10 squares.

Original entry on oeis.org

1, 20, 180, 960, 3380, 8424, 16320, 28800, 52020, 88660, 129064, 175680, 262080, 386920, 489600, 600960, 840500, 1137960, 1330420, 1563840, 2050344, 2611200, 2986560, 3358080, 4194240, 5318268, 5878440, 6299520, 7862400, 9619560

Offset: 0

N. J. A. Sloane

			G.f. = 1 + 20*x + 180*x^2 + 960*x^3 + 3380*x^4 + 8424*x^5 + 16320*x^6 + ...

E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 121.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 314.
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Chelsea Publishing Company, New York 1959, p. 135 section 9.3. MR0106147 (21 #4881)

T. D. Noe, Table of n, a(n) for n = 0..10000
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.
Shi-Chao Chen, Congruences for rs(n), Journal of Number Theory, Volume 130, Issue 9, September 2010, Pages 2028-2032.
J. Liouville, Nombre des représentations d’un entier quelconque sous la forme d’une somme de dix carrés, Journal de mathématiques pures et appliquées 2e série, tome 11 (1866), p. 1-8.
Index entries for sequences related to sums of squares

Row d=10 of A122141 and of A319574, 10th column of A286815.

(sum(x^(m^2),m=-10..10))^10;
# Alternative:
A000144list := proc(len) series(JacobiTheta3(0, x)^10, x, len+1);
seq(coeff(%, x, j), j=0..len-1) end: A000144list(30); # Peter Luschny, Oct 02 2018

Table[SquaresR[10, n], {n, 0, 30}] (* Ray Chandler, Jun 29 2008; updated by T. D. Noe, Jan 23 2012 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^10, {q, 0, n}]; (* Michael Somos, Aug 26 2015 *)
nmax = 50; CoefficientList[Series[Product[(1 - x^k)^10 * (1 + x^k)^30 / (1 + x^(2*k))^20, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 24 2017 *)

{a(n) = if( n<0, 0, polcoeff( sum(k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n))^10, n))}; /* Michael Somos, Sep 12 2005 */

Q = DiagonalQuadraticForm(ZZ, [1]*10)
Q.representation_number_list(37) # Peter Luschny, Jun 20 2014

Euler transform of period 4 sequence [ 20, -30, 20, -10, ...]. - Michael Somos, Sep 12 2005
Expansion of eta(q^2)^50 / (eta(q) * eta(q^4))^20 in powers of q. - Michael Somos, Sep 12 2005
a(n) = 4/5 * (A050456(n) + 16*A050468(n) + 8*A030212(n)) if n>0. - Michael Somos, Sep 12 2005
a(n) = (20/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - Seiichi Manyama, May 27 2017

Extended by Ray Chandler, Nov 28 2006

A000145 Number of ways of writing n as a sum of 12 squares.

Original entry on oeis.org

1, 24, 264, 1760, 7944, 25872, 64416, 133056, 253704, 472760, 825264, 1297056, 1938336, 2963664, 4437312, 6091584, 8118024, 11368368, 15653352, 19822176, 24832944, 32826112, 42517728, 51425088, 61903776, 78146664, 98021616

Offset: 0

N. J. A. Sloane

Apparently 8 | a(n). - Alexander R. Povolotsky, Oct 01 2011

			G.f. = 1 + 24*x + 264*x^2 + 1760*x^3 + 7944*x^4 + 25872*x^5 + 64416*x^6 + 133056*x^7 + ...

E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 121.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 314.

T. D. Noe, Table of n, a(n) for n = 0..10000
Shi-Chao Chen, Congruences for rs(n), Journal of Number Theory, Volume 130, Issue 9, September 2010, Pages 2028-2032.
Index entries for sequences related to sums of squares

Row d=12 of A122141 and of A319574, 12th column of A286815.

Cf. A000735, A029751.

A := Basis( ModularForms( Gamma0(4), 6), 25); A[1] + 24*A[2] + 264*A[3] + 1760*A[4]; /* Michael Somos, Aug 15 2015 */

(sum(x^(m^2),m=-10..10))^12; # gives g.f. for first 100 terms
t1:=(sum(x^(m^2), m=-n..n))^12; t2:=series(t1,x,n+1); t2[n+1]; # N. J. A. Sloane, Oct 01 2011
A000145list := proc(len) series(JacobiTheta3(0, x)^12, x, len+1);
seq(coeff(%, x, j), j=0..len-1) end: A000145list(27); # Peter Luschny, Oct 02 2018

SquaresR[12,Range[0,30]] (* Harvey P. Dale, Sep 07 2012 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^12, {q, 0, n}]; (* Michael Somos, Aug 15 2015 *)
nmax = 30; CoefficientList[Series[Product[(1 - x^(2*k))^12 * (1 + x^(2*k - 1))^24, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 10 2018 *)

{a(n) = if( n<0, 0, polcoeff( sum( k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n))^12, n))}; /* Michael Somos, Sep 21 2005 */

Expansion of eta(q^2)^60 / (eta(q) * eta(q^4))^24 in powers of q.
Euler transform of period 4 sequence [24, -36, 24, -12, ...]. - Michael Somos, Sep 21 2005
G.f.: (Sum_k x^k^2)^12 = theta_3(q)^12.
a(n) = A029751(n) + 16 * A000735(n). - Michael Somos, Sep 21 2005
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 64 (t/i)^6 f(t) where q = exp(2 Pi i t).
a(n) = (24/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - Seiichi Manyama, May 27 2017

A008453 Number of ways of writing n as a sum of 11 squares.

Original entry on oeis.org

1, 22, 220, 1320, 5302, 15224, 33528, 63360, 116380, 209550, 339064, 491768, 719400, 1095160, 1538416, 1964160, 2624182, 3696880, 4763220, 5686648, 7217144, 9528816, 11676280, 13495680, 16317048, 20787470, 25022184, 27785120, 32503680

Offset: 0

N. J. A. Sloane

nonn

E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 121.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 314.

T. D. Noe, Table of n, a(n) for n = 0..10000
Shi-Chao Chen, Congruences for rs(n), Journal of Number Theory, Volume 130, Issue 9, September 2010, Pages 2028-2032.
Shaun Cooper, On the number of representations of certain integers as sums of 11 or 13 squares, J. Number Theory 103 (2) (2003) 135-162
Index entries for sequences related to sums of squares

Row d=11 of A122141 and of A319574, 11th column of A286815.

Cf. A022042.

(sum(x^(m^2),m=-10..10))^11;
# Alternative:
A008453list := proc(len) series(JacobiTheta3(0, x)^11, x, len+1);
seq(coeff(%, x, j), j=0..len-1) end: A008453list(29); # Peter Luschny, Oct 02 2018

Table[SquaresR[11, n], {n, 0, 28}] (* Ray Chandler, Nov 28 2006 *)

G.f.: theta_3(0,q)^11, where theta_3 is the 3rd Jacobi theta function. - Ilya Gutkovskiy, Jan 13 2017

a(n) = (22/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - Seiichi Manyama, May 27 2017

Extended by Ray Chandler, Nov 28 2006

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

A185152 Expansion of (q/2) * phi(q)^3 (d/dq) phi(q) in powers of q.

Original entry on oeis.org

1, 6, 12, 12, 30, 72, 56, 24, 117, 180, 132, 144, 182, 336, 360, 48, 306, 702, 380, 360, 672, 792, 552, 288, 775, 1092, 1080, 672, 870, 2160, 992, 96, 1584, 1836, 1680, 1404, 1406, 2280, 2184, 720, 1722, 4032, 1892, 1584, 3510, 3312, 2256, 576, 2793, 4650

Offset: 1

Michael Somos, Jan 23 2012

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

Ramanujan's Eisenstein series: P(q) (see A006352), Q(q) (A004009), R(q) (A013973).

			G.f. = x + 6*x^2 + 12*x^3 + 28*x^4 + 30*x^5 + 72*x^6 + 56*x^7 + 24*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A000027, A000118, A000290, A046897, A186690, A222171.

Cf. A000122, A000700, A004009, A006352, A010054, A013973, A121373.

a[ n_] := If[ n == 0, 0, n Sum[ d Sign@Mod[d, 4], {d, Divisors@n}]]; (* Michael Somos, Jun 20 2015 *)
a[ n_] := Sign[n] With[ {f = Series[ EllipticTheta[ 3, 0, q], {q, 0, Abs@n + 1}]}, SeriesCoefficient[ (q/2) f^3 D[f, q], Abs@n]]; (* Michael Somos, Jun 20 2015 *)
a[ n_] := Sign[n] With[ {f = Series[ EllipticTheta[ 3, x, q], {q, 0, Abs@n + 1}]}, SeriesCoefficient[ (-1/8) f^3 D[f, x, x] /. x -> 0, Abs@n]]; (* Michael Somos, Jun 20 2015 *)

{a(n) = if( n==0, 0, n * sumdiv( n, d, if( d%4, d)))};

Expansion of (-1/8) * theta_3(0,q)^3 * theta_3(0,q)'' in powers of nome q.
Expansion of (-1/24) * q * (d/dq) (P(q) - 4 * P(q^4)) where P() is a Ramanujan Eisenstein series.
Expansion of (1/8) * (E(k^2) - (1-k^2) * K(k^2)) * K(k^2)^3 / (Pi/2)^4 in powers of nome q.
Multiplicative with a(2^e) = 3 * 2^e if e>0, a(p^e) = p^e * (p^(e+1) - 1) / (p - 1) otherwise.
G.f.: Sum_{k>0} k^2 * x^k / (1 + (-x)^k)^2.
G.f.: Sum_{k>0} k^2 * x^k / (1 - x^k)^2 * (mod(k, 4) > 0).
a(n) = n * Sum of divisors of n that are not divisible by 4 = n * A046897(n).
a(n) = - a(-n). for all n in Z. Convolution of A000118 and A186690.
From Amiram Eldar, Sep 12 2023: (Start)
Dirichlet g.f.: (1 - 1/2^(2*s-4)) * zeta(s-2) * zeta(s-1).
Sum_{k=1..n} a(k) ~ c * n^3, where c = Pi^2/24 = 0.41123... (A222171) . (End)

A000156 Number of ways of writing n as a sum of 24 squares.

Original entry on oeis.org

1, 48, 1104, 16192, 170064, 1362336, 8662720, 44981376, 195082320, 721175536, 2319457632, 6631997376, 17231109824, 41469483552, 93703589760, 200343312768, 407488018512, 793229226336, 1487286966928, 2697825744960, 4744779429216

Offset: 0

N. J. A. Sloane

The Carlitz paper has the wrong formula on p. 505, eq. (3). The factor in front of tau(n/2) should be -2^16 (not -2^12). The mistake appeared in the previous equation derived from eq. (2) where theta_3^(24) * 256*k^4*k'^4 was replaced by 2^8*g(q^2) which produces the factor 2^8*256 = 2^16. (One should subtract on p. 504 the second equation in the middle from the negative of the first equation. There is also a sign mistake in the sum term of the third equation from the bottom.) - Wolfdieter Lang, Sep 24 2016

Avner Ash and Robert Gross, Summing it up, Princeton University Press, 2016, p. 195, eq. (15.1).
E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 107.
G. H. Hardy, Ramanujan, 1940, Cambridge, reprinted with additional corrections and comments by AMS Chelsea Publishing, 1999, 2002, Providence, Rhode Island, ch. IX., pp. 153-155.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 314.

T. D. Noe, Table of n, a(n) for n = 0..10000
L. Carlitz, On the representation of an integer as the sum of twenty-four squares, Indagationes Mathematicae (Proceedings), 58 (1955) 504-506.
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.
Shi-Chao Chen, Congruences for rs(n), Journal of Number Theory, Volume 130, Issue 9, September 2010, Pages 2028-2032.
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.
Index entries for sequences related to sums of squares

Row d=24 of A122141 and of A319574, 24th column of A286815.

(sum(x^(m^2),m=-10..10))^24; seq(coeff(%,x,n), n=0..30);
# Alternative:
A000156list := proc(len) series(JacobiTheta3(0, x)^24, x, len+1);
seq(coeff(%, x, j), j=0..len-1) end: A000156list(21); # Peter Luschny, Oct 02 2018

Table[SquaresR[24, n], {n, 0, 20}] (* Ray Chandler, Nov 28 2006 *)

first(n)=my(x='x); x+=O(x^(n+1)); Vec((2*sum(k=1,sqrtint(n),x^k^2) + 1)^24) \\ Charles R Greathouse IV, Jul 29 2016

From Wolfdieter Lang, Sep 24 2016: (Start)
For n >= 1: a(n) = (16*sigma^*{11} - 128*(512*tau(n/2) + (-1)^n*259*tau(n)))/691, with sigma^*{11} = sigma_{11}^{e}(n) - sigma_{11}^{o}(n) if n even and sigma_{11}(n) otherwise. Here sigma_{11}(n) = A013959(n) and 0 if n is not an integer, sigma_{11}^{e}(n) and sigma_{11}^{o}(n) are the sums of the 11th power of the odd and even positive divisors of n, respectively. Ramanujan's tau(n) = A000594(n) and 0 if n is not an integer. This is from Hardy, ch. IX., p. 155, eqs. (9.17.1) and (9.17.2), and p.142 for the definition of sigma^*_{nu}(n). See also the Ash and Gross reference.
Another version, see the corrected Carlitz reference:
a(n) = (2^4*(sigma_{11}(n)- 2*sigma_{11}(n/2) + 2^{12}*sigma_{11}(n/4)) - 2^7*259*(-1)^n*tau(n) - 2^16*tau(n/2))/691, n >= 1.
(End)
a(n) = (48/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - Seiichi Manyama, May 27 2017

Extended by Ray Chandler, Nov 28 2006

A015680 Expansion of e.g.f. theta_3^(-1/2).

Original entry on oeis.org

1, -1, 3, -15, 81, -585, 4995, -46935, 499905, -6109425, 79791075, -1138096575, 17774982225, -294439570425, 5240530570275, -100050497922375, 2002010508122625, -42495475420022625, 954152290944727875

Offset: 0

N. J. A. Sloane

sign

			theta_3(q)^(-1/2) = 1 - q + 3/2 * q^2 - 5/2 * q^3 + 27/8 *q^4 - 39/8 * q^5 + ... = 1 - q + 3/2! * q^2 - 15/3! * q^3 + 81/4! * q^4 - 585/5! * q^5 + ... .

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.

Seiichi Manyama, Table of n, a(n) for n = 0..443

Cf. A000122 (theta_3), A015664, A186690.

Cf. A015682, A015683, A015684, A015685, A015687, A015690, A015691, A015693, A015694, A015695, A015696, A015697.

nmax = 25; CoefficientList[Series[EllipticTheta[3, 0, x]^(-1/2), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 23 2018 *)

a(0) = 1; a(n) = -(n-1)! * Sum_{k=1..n} A186690(k) * a(n-k)/(n-k)!. - Seiichi Manyama, Jul 07 2023

A186111 a(n) = -n if n odd, a(2n) = 3n if n odd, a(4n) = 2n.

Original entry on oeis.org

1, -3, 3, -2, 5, -9, 7, -4, 9, -15, 11, -6, 13, -21, 15, -8, 17, -27, 19, -10, 21, -33, 23, -12, 25, -39, 27, -14, 29, -45, 31, -16, 33, -51, 35, -18, 37, -57, 39, -20, 41, -63, 43, -22, 45, -69, 47, -24, 49, -75, 51, -26, 53, -81, 55, -28, 57, -87, 59, -30, 61, -93, 63

Offset: 1

Michael Somos, Feb 13 2011

			G.f. = x - 3*x^2 + 3*x^3 - 2*x^4 + 5*x^5 - 9*x^6 + 7*x^7 - 4*x^8 + 9*x^9 + ...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Michael Somos, Rational Function Multiplicative Coefficients.
Index entries for linear recurrences with constant coefficients, signature (-2,-3,-4,-3,-2,-1).

Cf. A068073, A186690, A186813.

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1-x)^3*(1-x^3)/(1-x^4)^2)); // G. C. Greubel, Aug 14 2018

Rest[CoefficientList[Series[x (1-x)^3(1-x^3)/(1-x^4)^2,{x,0,70}],x]] (* or *) LinearRecurrence[{-2,-3,-4,-3,-2,-1},{1,-3,3,-2,5,-9},70] (* Harvey P. Dale, Aug 08 2012 *)
a[ n_] := n If[ OddQ[n], 1, -(Mod[n/2, 2] + 1/2)]; (* Michael Somos, Apr 25 2015 *)
a[ n_] := n {1, -3/2, 1, -1/2}[[Mod[n, 4, 1]]]; (* Michael Somos, Apr 25 2015 *)

{a(n) = -(-1)^n * n * [1, 2, 3, 2] [n%4 + 1] / 2};

{a(n) = sign(n) * polcoeff( x * (1 - x)^3 * (1 - x^3) / (1 - x^4)^2 + x * O(x^abs(n)), abs(n))};

{a(n) = n * if( n%2, 1, -(n/2%2 + 1/2))}; /* Michael Somos, Apr 25 2015 */

a(n) is multiplicative with a(2) = -3, a(2^e) = -(2^(e-1)) if e>1, a(p^e) = p^e if p>2.
Euler transform of length 4 sequence [-3, 0, -1, 2].
G.f.: x * (1 - x)^3 * (1 - x^3) / (1 - x^4)^2.
G.f.: x * (1 + x + x^2) * (1 - x)^2 / ((1 + x)^2 * (1 + x^2)^2).
Dirichlet g.f. zeta(s-1)*( 1-5*2^(-s)+4^(1-s)). - R. J. Mathar, Mar 31 2011
a(n) = (-1)^(n+1)*n + (-1)^floor(n/2)*A027656(n-2). - R. J. Mathar, Mar 31 2011
a(n) = -2*a(n-1) - 3*a(n-2) - 4*a(n-3) - 3*a(n-4) - 2*a(n-5) - a(n-6) with a(1)=1, a(2)=-3, a(3)=3, a(4)=-2, a(5)=5, a(6)=-9. - Harvey P. Dale, Aug 08 2012
G.f.: 1/(1+x) - 1/(1+x)^2 - 1/(1+x^2) + 1/(1+x^2)^2. - Michael Somos, Apr 24 2015
a(n) = -a(-n) for all n in Z. - Michael Somos, Apr 24 2015
G.f.: f(x) - f(x^2) where f(x) := x / (1 + x)^2. - Michael Somos, May 07 2015
Moebius transform of A186690. - Michael Somos, Apr 25 2015
a(n) = -(-1)^n * A186813(n). - Michael Somos, May 07 2015
a(n) = n*cos(n*Pi/2)/2-n*(-1)^n. - Wesley Ivan Hurt, May 05 2021

A302855 Expansion of ((1 + 2 * Sum_{k>=1} q^(k^2))^16 - 1) / 32.

Original entry on oeis.org

1, 15, 140, 911, 4398, 16436, 49400, 124815, 279557, 582066, 1147332, 2124340, 3692390, 6160008, 10052904, 15910799, 24151410, 35732555, 52543100, 75891378, 106006432, 145588860, 200348520, 272046644, 359002903, 468778746, 615548600, 799793800, 1014602070, 1277001048

Offset: 1

Seiichi Manyama, Apr 14 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Expansion of ((1 + 2 * Sum_{k>=1} q^(k^2))^(2^m) - 1) / 2^(m + 1): A002654 (m=1), A046897 (m=2), A008457 (m=3), this sequence (m=4), A302857 (m=5).

Cf. A000152, A186690.

a(n) = A000152(n) / 32.

cp's OEIS Frontend

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A000156 Number of ways of writing n as a sum of 24 squares.