A122141 -id:A122141 - cp's OEIS frontend

A066536 Number of ways of writing n as a sum of n+1 squares.

1, 4, 12, 32, 90, 312, 1288, 5504, 22608, 88660, 339064, 1297056, 5043376, 19975256, 80027280, 321692928, 1291650786, 5177295432, 20748447108, 83279292960, 335056780464, 1351064867328, 5456890474248, 22063059606912

Offset: 0

Peter Bertok (peter(AT)bertok.com), Jan 07 2002

nonn

			There are a(2)=12 solutions (x,y,z) of 2=x^2+y^2+z^2: 3 permutations of (1,1,0), 3 permutations of (-1,-1,0) and 6 permutations of (1, -1,0).

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A004018, A005875, A000118, A066535.

Cf. A122141, A166952. - Paul D. Hanna, Oct 26 2009

Join[{1}, Table[SquaresR[n+1, n], {n, 24}]]
(* Calculation of constants {d,c}: *) {1/r, Sqrt[s/(2*Pi*r^2*Derivative[0, 0, 2][EllipticTheta][3, 0, r*s])]} /. FindRoot[{s == EllipticTheta[3, 0, r*s], r*Derivative[0, 0, 1][EllipticTheta][3, 0, r*s] == 1}, {r, 1/4}, {s, 5/2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Nov 16 2023 *)

{a(n)=local(THETA3=1+2*sum(k=1,sqrtint(n),x^(k^2))+x*O(x^n));polcoeff(THETA3^(n+1), n)} /* Paul D. Hanna, Oct 26 2009*/

a(n) equals the coefficient of x^n in the (n+1)-th power of Jacobi theta_3(x) where theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2). - Paul D. Hanna, Oct 26 2009
a(n) is divisible by n+1: a(n)/(n+1) = A166952(n) for n>=0. - Paul D. Hanna, Oct 26 2009
a(n) ~ c * d^n / sqrt(n), where d = 4.13273137623493996302796465... (= 1/radius of convergence A166952), c = 0.70710538549959357505200... . - Vaclav Kotesovec, Sep 10 2014

Edited by Dean Hickerson, Jan 12 2002
a(0) added by Paul D. Hanna, Oct 26 2009
Edited by R. J. Mathar, Oct 29 2009

A288515 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} ((1 + x^j)/(1 - x^j))^k.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 4, 0, 1, 6, 12, 8, 0, 1, 8, 24, 32, 14, 0, 1, 10, 40, 80, 76, 24, 0, 1, 12, 60, 160, 234, 168, 40, 0, 1, 14, 84, 280, 552, 624, 352, 64, 0, 1, 16, 112, 448, 1110, 1712, 1552, 704, 100, 0, 1, 18, 144, 672, 2004, 3912, 4896, 3648, 1356, 154, 0, 1, 20, 180, 960, 3346, 7896, 12600, 13120, 8184, 2532, 232, 0

Offset: 0

Ilya Gutkovskiy, Jun 10 2017

			Square array begins:
1,   1,    1,    1,     1,     1,  ...
0,   2,    4,    6,     8,    10,  ...
0,   4,   12,   24,    40,    60,  ...
0,   8,   32,   80,   160,   280,  ...
0,  14,   76,  234,   552,  1110,  ...
0,  24,  168,  624,  1712,  3913,  ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Index entries for sequences related to partitions

Columns k=0-24 give: A000007, A015128, A001934, A004404 (alternating values), A284286, A004406-A004425 (alternating values).
Rows n=0-2 give: A000012, A005843, A046092.
Main diagonal gives A270919.
Antidiagonal sums give A299108.
Cf. A122141, A286815.

# JacobiTheta4 is defined in A002448.
A288515Column(k, len) = JacobiTheta4(len, -k)
for k in 0:8 A288515Column(k, 8) |> println end # Peter Luschny, Mar 12 2018

Table[Function[k, SeriesCoefficient[Product[((1 + x^i)/(1 - x^i))^k, {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[1/EllipticTheta[4, 0, x]^k, {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

G.f. of column k: Product_{j>=1} ((1 + x^j)/(1 - x^j))^k.
G.f. of column k: 1/theta_4(x)^k, where theta_4() is the Jacobi theta function.
For asymptotics of column k see comment from Vaclav Kotesovec in A001934.

A319575 a(n) = (2/3)n(n^3 - 6n^2 + 11n - 3).

Original entry on oeis.org

0, 2, 4, 6, 24, 90, 252, 574, 1136, 2034, 3380, 5302, 7944, 11466, 16044, 21870, 29152, 38114, 48996, 62054, 77560, 95802, 117084, 141726, 170064, 202450, 239252, 280854, 327656, 380074, 438540, 503502, 575424, 654786, 742084, 837830, 942552, 1056794, 1181116

Offset: 0

Peter Luschny, Oct 01 2018

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000012 (m=0), A005843 (m=1), A046092 (m=2), A130809 (m=3), this sequence (m=4), A319576 (m=5), A319577 (m=6).
Column n=4 of A122141.
Cf. A319574.

a := n -> (2/3)*n*(n^3 - 6*n^2 + 11*n - 3):
seq(a(n), n=0..38);

A319575[n_] := 2/3*n*(n^3-6*n^2+11*n-3); Array[A319575, 50, 0] (* or *)
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 2, 4, 6, 24}, 50] (* Paolo Xausa, Feb 20 2024 *)

concat(0, Vec(2*x*(1 + x)*(1 - 4*x + 7*x^2) / (1 - x)^5 + O(x^40))) \\ Colin Barker, Oct 02 2018

a(n) = [x^4] JacobiTheta3(x)^n.
a(n) = A319574(n,4).
From Colin Barker, Oct 02 2018: (Start)
G.f.: 2*x*(1 + x)*(1 - 4*x + 7*x^2) / (1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>4. (End)

A319576 a(n) = (4/15)n(n - 1)(n^3 - 9n^2 + 26*n - 9).

Original entry on oeis.org

0, 0, 8, 24, 48, 112, 312, 840, 2016, 4320, 8424, 15224, 25872, 41808, 64792, 96936, 140736, 199104, 275400, 373464, 497648, 652848, 844536, 1078792, 1362336, 1702560, 2107560, 2586168, 3147984, 3803408, 4563672, 5440872, 6448000, 7598976, 8908680, 10392984

Offset: 0

Peter Luschny, Oct 01 2018

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000012 (m=0), A005843 (m=1), A046092 (m=2), A130809 (m=3), A319575 (m=4), this sequence (m=5), A319577 (m=6).
Column n=5 of A122141.
Cf. A319574.

a := n -> (4/15)*n*(n - 1)*(n^3 - 9*n^2 + 26*n - 9):
seq(a(n), n=0..41);

A319576[n_] := 4/15*n*(n-1)*(n^3-9*n^2+26*n-9); Array[A319576, 50, 0] (* or *)
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 0, 8, 24, 48, 112}, 50] (* Paolo Xausa, Feb 20 2024 *)

concat([0,0], Vec(8*x^2*(1 - 3*x + 3*x^2 + 3*x^3) / (1 - x)^6 + O(x^40))) \\ Colin Barker, Oct 02 2018

a(n) = [x^5] JacobiTheta3(x)^n.
a(n) = A319574(n,5).
From Colin Barker, Oct 02 2018: (Start)
G.f.: 8*x^2*(1 - 3*x + 3*x^2 + 3*x^3) / (1 - x)^6.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>5.
(End)

A319577 a(n) = (4/45)n(n - 2)(n - 1)(n^3 - 12n^2 + 47n - 15).

Original entry on oeis.org

0, 0, 0, 24, 96, 240, 544, 1288, 3136, 7392, 16320, 33528, 64416, 116688, 200928, 331240, 525952, 808384, 1207680, 1759704, 2508000, 3504816, 4812192, 6503112, 8662720, 11389600, 14797120, 19014840, 24189984, 30488976, 38099040, 47229864, 58115328, 71015296

Offset: 0

Peter Luschny, Oct 01 2018

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000012 (m=0), A005843 (m=1), A046092 (m=2), A130809 (m=3), A319575 (m=4), A319576 (m=5), this sequence (m=6).
Column n=6 of A122141.
Cf. A319574.

a := n -> (4/45)*n*(n - 2)*(n - 1)*(n^3 - 12*n^2 + 47*n - 15):
seq(a(n), n=0..41);

A319577[n_]:=4/45*n*(n-2)*(n-1)*(n^3-12*n^2+47*n-15); Array[A319577, 50, 0] (*or*)
LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 0, 0, 24, 96, 240, 544}, 50] (* Paolo Xausa, Feb 20 2024 *)

concat([0,0,0], Vec(8*x^3*(3 - 9*x + 9*x^2 + 5*x^3) / (1 - x)^7 + O(x^40))) \\ Colin Barker, Oct 02 2018

a(n) = [x^6] JacobiTheta3(x)^n.
a(n) = A319574(n,6).
From Colin Barker, Oct 02 2018: (Start)
G.f.: 8*x^3*(3 - 9*x + 9*x^2 + 5*x^3) / (1 - x)^7.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>6.
(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

A290430 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of (Sum_{j>=0} x^(j(j+1)(2*j+1)/6))^k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 1, 0, 0, 1, 4, 3, 0, 0, 0, 1, 5, 6, 1, 0, 1, 0, 1, 6, 10, 4, 0, 2, 0, 0, 1, 7, 15, 10, 1, 3, 2, 0, 0, 1, 8, 21, 20, 5, 4, 6, 0, 0, 0, 1, 9, 28, 35, 15, 6, 12, 3, 0, 0, 0, 1, 10, 36, 56, 35, 12, 20, 12, 0, 0, 0, 0, 1, 11, 45, 84, 70, 28, 31, 30, 4, 0, 1, 0, 0, 1, 12, 55, 120, 126, 64, 49, 60, 20, 0, 3, 0, 0, 0

Offset: 0

Ilya Gutkovskiy, Jul 31 2017

A(n,k) is the number of ways of writing n as a sum of k square pyramidal numbers (A000330).

			Square array begins:
1,  1,  1,  1,  1,   1,  ...
0,  1,  2,  3,  4,   5,  ...
0,  0,  1,  3,  6,  10,  ...
0,  0,  0,  1,  4,  10,  ...
0,  0,  0,  0,  1,   5,  ...
0,  1,  2,  3,  4,   6,  ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Eric Weisstein's World of Mathematics, Square Pyramidal Number
Index to sequences related to pyramidal numbers

Cf. A000330, A045847, A122141, A286815, A290429.
Cf. A000007 (column 0), A253903 (column 1), A282173 (column 6).
Main diagonal gives A303172.

Table[Function[k, SeriesCoefficient[Sum[x^(i (i + 1) (2 i + 1)/6), {i, 0, n}]^k, {x, 0, n}]][j - n], {j, 0, 13}, {n, 0, j}] // Flatten

G.f. of column k: (Sum_{j>=0} x^(j*(j+1)*(2*j+1)/6))^k.

A297331 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of (theta_3(q^(1/2))^k + theta_4(q^(1/2))^k)/2.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 4, 2, 0, 1, 12, 4, 0, 0, 1, 24, 6, 0, 0, 0, 1, 40, 24, 24, 4, 0, 0, 1, 60, 90, 96, 12, 8, 0, 0, 1, 84, 252, 240, 24, 24, 0, 0, 0, 1, 112, 574, 544, 200, 144, 8, 0, 2, 0, 1, 144, 1136, 1288, 1020, 560, 96, 48, 4, 0, 0, 1, 180, 2034, 3136, 3444, 1560, 400, 192, 6, 4, 0, 0

Offset: 0

Ilya Gutkovskiy, Dec 28 2017

			Square array begins:
1,  1,  1,   1,    1,    1,  ...
0,  0,  4,  12,   24,   40,  ...
0,  2,  4,   6,   24,   90,  ...
0,  0,  0,  24,   96,  240,  ...
0,  0,  4,  12,   24,  200,  ...
0,  0,  8,  24,  144,  560,  ...

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, p. 118.

Columns k=0..32 give A000007, A089798 (absolute values), A004018, A004015, A004011, A005930, A008428, A008429, A008430, A008431, A008432, A022042, A022043, A022044, A022045, A022046, A022047, A022048, A022049, A022050, A022051, A022052, A022053, A022054, A022055, A022056, A022057, A022058, A022059, A022060, A022061, A022062, A022063.
Main diagonal gives A303333.
Cf. A122141, A286815.

Table[Function[k, SeriesCoefficient[(EllipticTheta[3, 0, q^(1/2)]^k + EllipticTheta[4, 0, q^(1/2)]^k)/2, {q, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

G.f. of column k: (theta_3(q^(1/2))^k + theta_4(q^(1/2))^k)/2, where theta_() is the Jacobi theta function.

cp's OEIS Frontend

A066536 Number of ways of writing n as a sum of n+1 squares.

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A288515 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} ((1 + x^j)/(1 - x^j))^k.

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A319575 a(n) = (2/3)*n*(n^3 - 6*n^2 + 11*n - 3).

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A319576 a(n) = (4/15)*n*(n - 1)*(n^3 - 9*n^2 + 26*n - 9).

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A319577 a(n) = (4/45)*n*(n - 2)*(n - 1)*(n^3 - 12*n^2 + 47*n - 15).

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Maple

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

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Maple

Mathematica

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Extensions

A290430 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of (Sum_{j>=0} x^(j*(j+1)*(2*j+1)/6))^k.

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A319575 a(n) = (2/3)n(n^3 - 6n^2 + 11n - 3).

A319576 a(n) = (4/15)n(n - 1)(n^3 - 9n^2 + 26*n - 9).

A319577 a(n) = (4/45)n(n - 2)(n - 1)(n^3 - 12n^2 + 47n - 15).

A290430 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of (Sum_{j>=0} x^(j(j+1)(2*j+1)/6))^k.