cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Original entry on oeis.org

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

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Author

N. J. A. Sloane

Keywords

References

  • 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).

Links

Crossrefs

Row d=9 of A122141 and of A319574, 9th column of A286815.
Cf. A008431.

Programs

  • Maple
    (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
  • Mathematica
    Table[SquaresR[9, n], {n, 0, 32}] (* Ray Chandler, Nov 28 2006 *)
  • Python
    # 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
  • Sage
    Q = DiagonalQuadraticForm(ZZ, [1]*9)
Q.representation_number_list(37) # Peter Luschny, Jun 20 2014

Formula

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

Extensions

Extended by Ray Chandler, Nov 28 2006

A008424 Theta series of {D_9}* lattice.

Original entry on oeis.org

1, 0, 0, 0, 18, 0, 0, 0, 144, 512, 0, 0, 672, 0, 0, 0, 2034, 4608, 0, 0, 4320, 0, 0, 0, 7392, 18432, 0, 0, 12672, 0, 0, 0, 22608, 47616, 0, 0, 34802, 0, 0, 0, 44640, 101376, 0, 0, 60768, 0, 0, 0, 93984, 193536, 0, 0, 125280, 0, 0, 0, 141120, 324096, 0, 0
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			G.f. = 1 + 18*q^4 + 144*q^8 + ...

References

  • J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 120.

Links

Crossrefs

Cf. A008431.

Programs

  • Magma
    L := Dual(Lattice("D", 9));
B := Basis(ThetaSeriesModularFormSpace(L), 100);
S := [ 1, 0, 0, 0, 18];
Coefficients(&+[B[i] * S[i] : i in [1..5]]); // Andy Huchala, Jul 24 2021
  • PARI
    N=66;  q='q+O('q^N);
T3(q) = eta(q^2)^5 / ( eta(q)^2 * eta(q^4)^2 );
T2(q) = eta(q^4)^2 / eta(q^2);
Vec( T3(q^4)^9 + (2 * q * T2(q^4))^9 )
\\ Joerg Arndt, Mar 29 2018

Formula

Theta series in terms of Jacobi theta series: (theta_2)^9 + (theta_3)^9. - Sean A. Irvine, Mar 28 2018

Extensions

More terms from Andy Huchala, Jul 24 2021

A008436 Theta series of {D_9}^{+} packing.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 144, 256, 0, 0, 0, 0, 0, 0, 2034, 2304, 0, 0, 0, 0, 0, 0, 7392, 9216, 0, 0, 0, 0, 0, 0, 22608, 23808, 0, 0, 0, 0, 0, 0, 44640, 50688, 0, 0, 0, 0, 0, 0, 93984, 96768, 0, 0, 0, 0, 0, 0, 141120
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			G.f.: 1 + 144*q^2 + 256*q^(9/4) + 2034*q^4 + 2304*q^(17/4) + ... .

References

  • J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 120.

Links

Crossrefs

Cf. A000122 (theta_3(q)), A002448 (theta_4(q)), A008431.

Formula

From Seiichi Manyama, Oct 21 2018: (Start)
Expansion of (theta_2(q)^9 + theta_3(q)^9 + theta_4(q)^9)/2 in powers of q^(1/4).
Expansion of (Sum_{k=-oo..oo} q^((k+1/2)^2))^9 + (Sum_{k=-oo..oo} q^(k^2))^9 + (Sum_{k=-oo..oo} (-1)^k * q^(k^2))^9 in powers of q^(1/4). (End)

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

Views

Author

Ilya Gutkovskiy, Dec 28 2017

Keywords

Examples

			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,  ...

Links

Crossrefs

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.

Programs

  • Mathematica
    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

Formula

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.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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