A173680 -id:A173680 - cp's OEIS frontend

A051343 Number of ways of writing n as a sum of 3 nonnegative cubes (counted naively).

1, 3, 3, 1, 0, 0, 0, 0, 3, 6, 3, 0, 0, 0, 0, 0, 3, 3, 0, 0, 0, 0, 0, 0, 1, 0, 0, 3, 6, 3, 0, 0, 0, 0, 0, 6, 6, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 3, 0, 0, 0, 0, 0, 0, 3, 0, 3, 6, 3, 0, 0, 0, 0, 0, 6, 6, 0, 0, 0, 0, 0, 0, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 6

Offset: 0

N. J. A. Sloane

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Cf. A051344.

Sums of k cubes, number of ways of writing n as, for k=1..9: A010057, A173677, A051343, A173678, A173679, A173680, A173676, A173681, A173682.

Maple
```
series(add(x^(n^3), n=0..10)^3,x,1000);
```

first(n)=my(s=vector(n+1)); for(k=0,sqrtnint(n,3), s[k^3+1]=1); Vec(Ser(s,,n+1)^3) \\ Charles R Greathouse IV, Sep 16 2016

A173677 Number of ways of writing n as a sum of two nonnegative cubes.

Original entry on oeis.org

1, 2, 1, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2

Offset: 0

N. J. A. Sloane, Nov 24 2010

Order matters. This is the coefficient of q^n in the expansion of {Sum_{m>=0} q^(m^3)}^2.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A004829, A008451, A010057, A051343, A173676.

Sums of k cubes, number of ways of writing n as, for k=1..9: A010057, A173677, A051343, A173678, A173679, A173680, A173676, A173681, A173682.

list(n)=my(q='q);Vec(sum(m=0,(n+.5)^(1/3),q^(m^3),O(q^(n+1)))^2) \\ Charles R Greathouse IV, Jun 07 2012

a(n) = Sum_{k=0..n} c(k) * c(n-k), where c = A010057. - Wesley Ivan Hurt, Nov 09 2023

A173682 Number of ways of writing n as a sum of 9 nonnegative cubes.

Original entry on oeis.org

1, 9, 36, 84, 126, 126, 84, 36, 18, 73, 252, 504, 630, 504, 252, 72, 45, 252, 756, 1260, 1260, 756, 252, 36, 84, 504, 1260, 1689, 1332, 756, 588, 630, 630, 882, 1332, 1341, 1134, 1638, 2520, 2520, 1638, 1008, 828, 756, 1638, 3780, 5040, 3780, 1596, 504, 252, 588, 2520, 5040, 5076, 2772, 1296, 1332, 1296, 1386, 2772, 3816, 2772, 2142, 3798, 5121, 4032

Offset: 0

N. J. A. Sloane, Nov 25 2010

nonn

Order matters. This is the coefficient of q^n in the expansion of {Sum_{m>=0} q^(m^3)}^9.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A173677.

Sums of k cubes, number of ways of writing n as, for k=1..9: A010057, A173677, A051343, A173678, A173679, A173680, A173676, A173681, A173682.

A173676 Number of ways of writing n as a sum of seven nonnegative cubes.

Original entry on oeis.org

1, 7, 21, 35, 35, 21, 7, 1, 7, 42, 105, 140, 105, 42, 7, 0, 21, 105, 210, 210, 105, 21, 0, 0, 35, 140, 210, 147, 77, 105, 140, 105, 77, 112, 105, 77, 210, 420, 420, 210, 63, 42, 21, 105, 420, 630, 420, 105, 7, 7, 0, 140, 420, 420, 161, 105, 211, 210, 105, 126, 210, 105, 105, 420, 637, 462, 210, 182, 147, 42, 217, 630, 672, 420, 420, 427, 210, 42

Offset: 0

N. J. A. Sloane, Nov 24 2010

Order matters. This is the coefficient of q^n in the expansion of {Sum_{m>=0} q^(m^3)}^7.

It is known that a(n)>0 if n is even and > 454.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
N. D. Elkies, Every even number greater than 454 is the sum of seven cubes, arXiv:1009.3983

Cf. A004829, A008451.

Sums of k cubes, number of ways of writing n as, for k=1..9: A010057, A173677, A051343, A173678, A173679, A173680, A173676, A173681, A173682.

A173678 Number of ways of writing n as a sum of 4 nonnegative cubes.

Original entry on oeis.org

1, 4, 6, 4, 1, 0, 0, 0, 4, 12, 12, 4, 0, 0, 0, 0, 6, 12, 6, 0, 0, 0, 0, 0, 4, 4, 0, 4, 12, 12, 4, 0, 1, 0, 0, 12, 24, 12, 0, 0, 0, 0, 0, 12, 12, 0, 0, 0, 0, 0, 0, 4, 0, 0, 6, 12, 6, 0, 0, 0, 0, 0, 12, 12, 4, 12, 12, 4, 0, 0, 6, 0, 12, 24, 12, 0, 0, 0, 0, 0, 12, 16, 4, 0, 0, 0, 0, 0, 4, 4, 0, 12, 24, 12, 0, 0, 0, 0, 0, 24, 24, 0, 0, 0, 0, 0, 0, 12, 1, 0, 0

Offset: 0

N. J. A. Sloane, Nov 24 2010

Order matters. This is the coefficient of q^n in the expansion of {Sum_{m>=0} q^(m^3)}^4.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A004829, A008451, A010057, A173677, A051343, A173676.
Sums of k cubes, number of ways of writing n as, for k=1..9: A010057, A173677, A051343, A173678, A173679, A173680, A173676, A173681, A173682.
Without order you get A025448.

A173679 Number of ways of writing n as a sum of 5 nonnegative cubes.

Original entry on oeis.org

1, 5, 10, 10, 5, 1, 0, 0, 5, 20, 30, 20, 5, 0, 0, 0, 10, 30, 30, 10, 0, 0, 0, 0, 10, 20, 10, 5, 20, 30, 20, 5, 5, 5, 0, 20, 60, 60, 20, 0, 1, 0, 0, 30, 60, 30, 0, 0, 0, 0, 0, 20, 20, 0, 10, 30, 30, 10, 0, 5, 0, 0, 30, 60, 35, 20, 30, 20, 5, 0, 30, 30, 20, 60, 60, 20, 0, 0, 10, 0, 30, 70, 50, 10, 0, 0, 0, 0, 20, 40, 20, 20, 60, 60, 20, 0, 5, 10, 0, 60, 120

Offset: 0

N. J. A. Sloane, Nov 25 2010

nonn

Order matters. This is the coefficient of q^n in the expansion of {Sum_{m>=0} q^(m^3)}^5.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A173677.

Sums of k cubes, number of ways of writing n as, for k=1..9: A010057, A173677, A051343, A173678, A173679, A173680, A173676, A173681, A173682.

A173681 Number of ways of writing n as a sum of 8 nonnegative cubes.

Original entry on oeis.org

1, 8, 28, 56, 70, 56, 28, 8, 9, 56, 168, 280, 280, 168, 56, 8, 28, 168, 420, 560, 420, 168, 28, 0, 56, 280, 560, 568, 336, 224, 280, 280, 238, 336, 428, 336, 406, 840, 1120, 840, 392, 224, 168, 224, 840, 1680, 1680, 840, 196, 56, 28, 280, 1120, 1680, 1148, 448, 428, 568, 420, 448, 868, 840, 448, 840, 1689, 1736, 1008, 616, 616, 336, 476, 1688, 2576

Offset: 0

N. J. A. Sloane, Nov 25 2010

nonn

Order matters. This is the coefficient of q^n in the expansion of {Sum_{m>=0} q^(m^3)}^8.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
L. E. Dickson, All integers except 23 and 239 are sums of eight cubes, Bull. Amer. Math. Soc. 45 (1939), 588-591.

Cf. A004830, A173677.

Sums of k cubes, number of ways of writing n as, for k=1..9: A010057, A173677, A051343, A173678, A173679, A173680, A173676, A173681, A173682.

lista(n)=my(q='q); Vec(sum(m=0, (n+.5)^(1/3), q^(m^3), O(q^(n+1)))^8); \\ Michel Marcus, Apr 12 2016

A340979 Number of ways to write n as an ordered sum of 6 positive cubes.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 15, 0, 0, 0, 0, 0, 0, 20, 0, 0, 0, 0, 6, 0, 15, 0, 0, 0, 0, 30, 0, 6, 0, 0, 0, 0, 60, 0, 1, 0, 0, 0, 0, 60, 0, 0, 0, 0, 15, 0, 30, 0, 0, 0, 0, 60, 0, 6, 0, 6, 0, 0, 90, 0, 0, 0, 30, 0, 0, 60, 0, 0, 0, 60, 20, 0, 15, 0, 0, 0, 60, 60, 0, 0, 0, 30, 0, 30, 60

Offset: 6

Ilya Gutkovskiy, Feb 01 2021

nonn

Robert Israel, Table of n, a(n) for n = 6..10000
Index entries for sequences related to sums of cubes

Cf. A000578, A010057, A025459, A051344, A173680, A280618, A340977, A340978, A340980, A340981, A340982, A340983.

b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add((s->
      `if`(s>n, 0, b(n-s, t-1)))(j^3), j=1..iroot(n, 3))))
    end:
a:= n-> b(n, 6):
seq(a(n), n=6..98);  # Alois P. Heinz, Feb 01 2021

nmax = 98; CoefficientList[Series[Sum[x^(k^3), {k, 1, Floor[nmax^(1/3)] + 1}]^6, {x, 0, nmax}], x] // Drop[#, 6] &

G.f.: (Sum_{k>=1} x^(k^3))^6.

A290054 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of (Sum_{j>=0} x^(j^3))^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, 0, 0, 1, 6, 10, 4, 0, 0, 0, 0, 1, 7, 15, 10, 1, 0, 0, 0, 0, 1, 8, 21, 20, 5, 0, 0, 0, 1, 0, 1, 9, 28, 35, 15, 1, 0, 0, 2, 0, 0, 1, 10, 36, 56, 35, 6, 0, 0, 3, 2, 0, 0, 1, 11, 45, 84, 70, 21, 1, 0, 4, 6, 0, 0, 0

Offset: 0

Ilya Gutkovskiy, Jul 19 2017

A(n,k) is the number of ways of writing n as a sum of k nonnegative cubes.

			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,  0,  0,  0,  0,   1,  ...

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

Columns k=0-9 give: A000007, A010057, A173677, A051343, A173678, A173679, A173680, A173676, A173681, A173682.
Main diagonal gives A291700.
Antidiagonal sums give A302019.
Cf. A045847, A122141, A286815.

Table[Function[k, SeriesCoefficient[Sum[x^i^3, {i, 0, n}]^k, {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten

G.f. of column k: (Sum_{j>=0} x^(j^3))^k.

A004829 Numbers that are the sum of at most 7 positive cubes.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69

Offset: 1

N. J. A. Sloane

McCurley proves that every n > exp(exp(13.97)) is in A003330 and hence in this sequence. Siksek proves that all n > 454 are in this sequence. - Charles R Greathouse IV, Jun 29 2022

T. D. Noe, Table of n, a(n) for n = 1..1000
Jan Bohman and Carl-Erik Froberg, Numerical investigation of Waring's problem for cubes, Nordisk Tidskr. Informationsbehandling (BIT) 21 (1981), 118-122.
K. S. McCurley, An effective seven-cube theorem, J. Number Theory, 19 (1984), 176-183.
Samir Siksek, Every integer greater than 454 is the sum of at most seven positive cubes, Algebra and Number Theory 10:10 (2016), pp. 2093-2119.
Index entries for linear recurrences with constant coefficients, signature (2,-1).
Index entries for sequences related to sums of cubes

Complement of A018889; subsequence of A003330.
Sums of k cubes, number of ways of writing n as, for k=1..9: A010057, A173677, A051343, A173678, A173679, A173680, A173676, A173681, A173682.
Cf. A018888.

cp's OEIS Frontend

A051343 Number of ways of writing n as a sum of 3 nonnegative cubes (counted naively).

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A173677 Number of ways of writing n as a sum of two nonnegative cubes.

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A173682 Number of ways of writing n as a sum of 9 nonnegative cubes.

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A173676 Number of ways of writing n as a sum of seven nonnegative cubes.

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A173678 Number of ways of writing n as a sum of 4 nonnegative cubes.

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A173679 Number of ways of writing n as a sum of 5 nonnegative cubes.

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A173681 Number of ways of writing n as a sum of 8 nonnegative cubes.

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A340979 Number of ways to write n as an ordered sum of 6 positive cubes.

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A290054 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of (Sum_{j>=0} x^(j^3))^k.

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A004829 Numbers that are the sum of at most 7 positive cubes.

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