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.

A224215 Number of nonnegative solutions to x^3 + y^3 + z^3 <= n^3.

Original entry on oeis.org

1, 4, 11, 30, 66, 115, 200, 302, 441, 619, 829, 1085, 1395, 1771, 2200, 2666, 3228, 3843, 4564, 5351, 6185, 7143, 8158, 9349, 10526, 11934, 13375, 14896, 16652, 18381, 20370, 22411, 24629, 26963, 29406, 32101, 34840, 37766, 40920, 44164, 47587, 51200
Offset: 0

Views

Author

Alex Ratushnyak, Apr 01 2013

Keywords

Examples

			For n=1, the four solutions are {0,0,0}, {0,0,1}, {0,1,0} and {1,0,0}, so a(1)=4.

Links

Crossrefs

Cf. A224214.

Programs

  • PARI
    a(n) = n++; p = Pol((1/(1 - x))*sum(k=0, n, x^(k^3))^3 + O(x^(n^3))); polcoeff(p, (n-1)^3); \\ Michel Marcus, Apr 21 2018
  • PARI
    \\ See PARI link. David A. Corneth, May 22 2018
  • Python
    for a in range(99):
  n = a*a*a
  k = 0
  for x in range(99):
    s = x*x*x
    if s>n: break
    for y in range(99):
        sy = s + y*y*y
        if sy>n: break
        for z in range(99):
            sz = sy + z*z*z
            if sz>n: break
            k+=1
  print(k, end=',')

Formula

a(n) = [x^(n^3)] (1/(1 - x))*(Sum_{k>=0} x^(k^3))^3. - Ilya Gutkovskiy, Apr 20 2018

A303169 a(n) = [x^(n^3)] (1/(1 - x))*(Sum_{k>=0} x^(k^3))^n.

Original entry on oeis.org

1, 2, 6, 30, 241, 2093, 23059, 276056, 3657901, 51751598, 792918670, 13031054778, 228632547574, 4247832219975, 83138970732860, 1710953260292025, 36844216654753387, 827664913984323748, 19363023028132371129, 470436686367280495474, 11843579175327033093769
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 19 2018

Keywords

Comments

Number of nonnegative solutions to (x_1)^3 + (x_2)^3 + ... + (x_n)^3 <= n^3.

Links

Crossrefs

Main diagonal of A303484.
Cf. A000578, A010052, A224214, A224215, A290054, A298671, A302863.

Programs

  • Mathematica
    Table[SeriesCoefficient[1/(1 - x) Sum[x^k^3, {k, 0, n}]^n, {x, 0, n^3}], {n, 0, 20}]

A303484 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = [x^(n^3)] (1/(1 - x))*(Sum_{j>=0} x^(j^3))^k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 11, 11, 5, 1, 1, 6, 20, 30, 18, 6, 1, 1, 7, 37, 84, 66, 26, 7, 1, 1, 8, 70, 237, 241, 115, 37, 8, 1, 1, 9, 135, 662, 853, 500, 200, 50, 9, 1, 1, 10, 264, 1780, 2847, 2093, 1012, 302, 63, 10, 1, 1, 11, 520, 4536, 9033, 8451, 4914, 1769, 441, 80, 11, 1
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 24 2018

Keywords

Comments

A(n,k) is the number of nonnegative solutions to (x_1)^3 + (x_2)^3 + ... + (x_k)^3 <= n^3.

Examples

			Square array begins:
1,  1,   1,    1,    1,     1,  ...
1,  2,   3,    4,    5,     6,  ...
1,  3,   6,   11,   20,    37,  ...
1,  4,  11,   30,   84,   237,  ...
1,  5,  18,   66,  241,   853,  ...
1,  6,  26,  115,  500,  2093,  ...

Links

Crossrefs

Columns k=0..4 give A000012, A000027, A224214, A224215.
Main diagonal gives A303169.
Cf. A290054, A302998.

Programs

  • Mathematica
    Table[Function[k, SeriesCoefficient[1/(1 - x) Sum[x^i^3, {i, 0, n}]^k, {x, 0, n^3}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
Showing 1-3 of 3 results.

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