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.

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

A338932 Numbers k such that the Diophantine equation x^3 + y^3 + z^3 = k has nontrivial primitive parametric solutions.

Original entry on oeis.org

1, 2, 128, 729, 1458, 4096, 65536, 93312, 2985984, 3906250, 16777216, 28697814, 33554432, 47775744, 80707214, 244140625, 250000000, 387420489, 1836660096, 2847656250, 4715895382, 5165261696, 12230590464, 13841287201, 17179869184, 21208998746, 24461180928
Offset: 1

Views

Author

XU Pingya, Nov 16 2020

Keywords

Comments

The data are derived from the following formula:
(a^3 - 6*t^3)^3 + (a^3 + 6*t^3)^3 + (-6*a*t^2)^3 = 2*a^9;
(4*a^3 - 3*t^3)^3 + (4*a^3 + 3*t^3)^3 + (-6*a*t^2)^3 = 128*a^9 = 2*4^3*a^9;
(9*a^3 - 2*t^3)^3 + (9*a^3 + 2*t^3)^3 + (-6*a*t^2)^3 = 1458*a^9 = 2*9^3*a^9;
(36*a^3 - t^3)^3 + (36*a^3 + t^3)^3 + (-6*a*t^2)^3 = 93312*a^9 = 2*36^3*a^9;
((3*a^3)*t - 9*t^4)^3 + (9*t^4)^3 + (a^4 - 9*a*t^3)^3 = a^12;
((9*a^3)*t - t^4)^3 + (t^4)^3 + (9*a^4 - 3*a*t^3)^3 = 729*a^12 = 9^3*a^12.

Examples

			128 is a term, because (4 - 3*(2*n - 1)^3, 4 + 3*(2*n - 1)^3, -3*(2*n - 1)^2) is a nontrivial primitive parametric solution of x^3 + y^3 + z^3 = 128.

References

  • R. K. Guy, Unsolved Problems in Number Theory, D5.

Links

Crossrefs

Cf. A113490, A173515, A175365, A224215, A226903, A281224, A327586, A338933.

Programs

  • Mathematica
    t1 = 2*{1, 5, 7, 11, 13}^9;
t2 = 128*{1, 2, 4, 5, 7, 8}^9;
t3 = 1458*{1, 3, 5, 7, 9}^9;
t4 = 93312*{1, 2, 3, 4, 5}^9;
t5 = {1, 2, 4, 5, 7}^12;
t6 = 729*{1, 2, 3, 4, 5}^12;
Take[Union[t1, t2, t3, t4, t5, t6], 27]

A338933 Numbers k such that the Diophantine equation x^3 + y^3 + 2*z^3 = k has nontrivial primitive parametric solutions.

Original entry on oeis.org

2, 16, 128, 1024, 1458, 8192, 11664, 31250, 65536, 93312, 235298, 524288, 746496, 1062882, 2000000, 3543122, 3906250, 5971968, 9653618, 15059072, 22781250, 28697814, 33554432, 47775744, 48275138, 68024448, 80707214, 94091762, 128000000, 171532242, 226759808
Offset: 1

Views

Author

XU Pingya, Nov 16 2020

Keywords

Comments

The data are derived from the following formula:
(a^2 - a*t - t^2)^3 + (a^2 + a*t - t^2)^3 + 2*(t^2)^3 = 2*a^6
(a^3 - 3*t^3)^3 + (a^3 + 3*t^3) + 2*(-3*a*t^2)^3 = 2*a^9;
(9*a^3 - t^3)^3 + (9*a^3 + t^3)^3 + 2*(-3*a*t^2)^3 = 1458*a^9;
(6*a^3*t - 72*t^4)^3 + (72*t^4)^3 + 2*(a^4 - 36*a*t^3)^3 = 2*a^12;
(6*a^3*t - 9*t^4)^3 + (9*t^4)^3 + 2*(2*a^4 - 9*a*t^3)^3 = 16*a^12 = 2*2^3*a^12;
(18*a^3*t - 8*t^4)^3 + (8*t^4)^3 + 2*(9*a^4 - 12*a*t^3)^3 = 1458*a^12 = 2*9^3*a^12;
(18*a^3*t - t^4)^3 + (t^4)^3 + 2*(18*a^4 - 3*a*t^3)^3 = 11664*a^12 = 2*18^3*a^12.

Examples

			16 is a term, because when t is an integer, (6*(2*t + 1) - 9*(2*t + 1)^4, 9*(2*t + 1)^4, 2 - 9*(2*t + 1)^3) is a nontrivial primitive parametric solution of x^3 + y^3 + 2*z^3 = 16.

References

  • R. K. Guy, Unsolved Problems in Number Theory, D5.

Links

Crossrefs

Cf. A113490, A173515, A175365, A224215, A226903, A281224, A327586, A338932.

Programs

  • Mathematica
    t1 = 2*Range[23]^6;
t2 = 2*{1, 2, 4, 5, 7, 8}^9;
t3 = 1458*Range[4]^9;
t4 = 2*{1, 5}^12;
t5 = 16*{1, 2, 4}^12;
t6 = 1458*{1, 3}^12;
t7 = 11664*{1, 2, 3}^12;
Take[Union[t1, t2, t3, t4, t5, t6, t7], 31]

Extensions

Missing terms 1024 and 746496 added by XU Pingya, Mar 14 2022
Showing 1-4 of 4 results.

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