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 1-3 of 3 results.

A000152 Number of ways of writing n as a sum of 16 squares.

Original entry on oeis.org

1, 32, 480, 4480, 29152, 140736, 525952, 1580800, 3994080, 8945824, 18626112, 36714624, 67978880, 118156480, 197120256, 321692928, 509145568, 772845120, 1143441760, 1681379200, 2428524096, 3392205824, 4658843520, 6411152640
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 314.
  • E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 107.

Links

Crossrefs

Row d=16 of A122141 and of A319574, 16th column of A286815.
Cf. A022047(n) = A000152(2*n).

Programs

  • Maple
    (sum(x^(m^2),m=-10..10))^16;
# Alternative:
A000152list := proc(len) series(JacobiTheta3(0, x)^16, x, len+1);
seq(coeff(%, x, j), j=0..len-1) end: A000152list(24); # Peter Luschny, Oct 02 2018
  • Mathematica
    Table[SquaresR[16, n], {n, 0, 23}] (* Ray Chandler, Nov 28 2006 *)
CoefficientList[EllipticTheta[3, 0, x]^16 + O[x]^24, x] (* Jean-François Alcover, Jul 06 2017 *)
  • PARI
    first(n)=my(x='x); x+=O(x^(n+1)); Vec((2*sum(k=1,sqrtint(n),x^k^2) + 1)^16) \\ Charles R Greathouse IV, Jul 29 2016

Formula

G.f.: theta_3(0,q)^16, where theta_3 is the 3rd Jacobi theta function. - Ilya Gutkovskiy, Jan 13 2017
a(n) = (32/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

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.

A047803 Theta series of 24-dimensional lattice D_16 + E_8.

Original entry on oeis.org

1, 720, 146512, 8565952, 196434000, 2317602912, 17225174720, 93710019456, 407503812688, 1487330605840, 4744665673056, 13523554712640, 35251061710528, 84944383084512, 192127846616960, 409984044318336, 834655060080720
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Crossrefs

Cf. A004009, A022047

Formula

G.f.: d16(q) * e8(q) where e8(q) = 1 + 240*q^2 + 2160*q^4 + ... is the theta series for A004009 and d16(q) = 1 + 480*q^2 + 29152*q^4 + ... is the theta series for A022047. - Sean A. Irvine, May 21 2021

Extensions

Description corrected by Dave Yuen, May 01 2002
Showing 1-3 of 3 results.

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

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