A051344 -id:A051344 - 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

A340977 Number of ways to write n as an ordered sum of 4 positive cubes.

Original entry on oeis.org

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

Offset: 4

Ilya Gutkovskiy, Feb 01 2021

nonn

Robert Israel, Table of n, a(n) for n = 4..10000
Index entries for sequences related to sums of cubes

Cf. A000578, A010057, A025457, A051344, A173678, A280618, A340978, A340979, 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, 4):
seq(a(n), n=4..100);  # Alois P. Heinz, Feb 01 2021

nmax = 100; CoefficientList[Series[Sum[x^(k^3), {k, 1, Floor[nmax^(1/3)] + 1}]^4, {x, 0, nmax}], x] // Drop[#, 4] &

G.f.: (Sum_{k>=1} x^(k^3))^4.

A340978 Number of ways to write n as an ordered sum of 5 positive cubes.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 5, 0, 5, 0, 0, 0, 0, 20, 0, 1, 0, 0, 0, 0, 30, 0, 0, 0, 0, 0, 0, 20, 0, 0, 0, 0, 10, 0, 5, 0, 0, 0, 0, 30, 0, 0, 0, 5, 0, 0, 30, 0, 0, 0, 20, 0, 0, 10, 0, 0, 0, 30, 10, 0, 0, 0, 0, 0, 20, 20, 0, 0, 0, 20, 0, 5, 10

Offset: 5

Ilya Gutkovskiy, Feb 01 2021

nonn

Robert Israel, Table of n, a(n) for n = 5..10000
Index entries for sequences related to sums of cubes

Cf. A000578, A010057, A025458, A051344, A173679, A280618, A340977, A340979, 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, 5):
seq(a(n), n=5..97);  # Alois P. Heinz, Feb 01 2021

nmax = 97; CoefficientList[Series[Sum[x^(k^3), {k, 1, Floor[nmax^(1/3)] + 1}]^5, {x, 0, nmax}], x] // Drop[#, 5] &

G.f.: (Sum_{k>=1} x^(k^3))^5.

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.

A340980 Number of ways to write n as an ordered sum of 7 positive cubes.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 21, 0, 0, 0, 0, 0, 0, 35, 0, 0, 0, 0, 7, 0, 35, 0, 0, 0, 0, 42, 0, 21, 0, 0, 0, 0, 105, 0, 7, 0, 0, 0, 0, 140, 0, 1, 0, 0, 21, 0, 105, 0, 0, 0, 0, 105, 0, 42, 0, 7, 0, 0, 210, 0, 7, 0, 42, 0, 0, 210, 0, 0, 0, 105, 35, 0, 105, 0, 0, 0, 140, 140, 0, 21, 0, 42

Offset: 7

Ilya Gutkovskiy, Feb 01 2021

nonn

Robert Israel, Table of n, a(n) for n = 7..10000
Index entries for sequences related to sums of cubes

Cf. A000578, A010057, A025460, A051344, A173676, A280618, A340977, A340978, A340979, 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, 7):
seq(a(n), n=7..96);  # Alois P. Heinz, Feb 01 2021

nmax = 96; CoefficientList[Series[Sum[x^(k^3), {k, 1, Floor[nmax^(1/3)] + 1}]^7, {x, 0, nmax}], x] // Drop[#, 7] &

G.f.: (Sum_{k>=1} x^(k^3))^7.

A340981 Number of ways to write n as an ordered sum of 8 positive cubes.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 28, 0, 0, 0, 0, 0, 0, 56, 0, 0, 0, 0, 8, 0, 70, 0, 0, 0, 0, 56, 0, 56, 0, 0, 0, 0, 168, 0, 28, 0, 0, 0, 0, 280, 0, 8, 0, 0, 28, 0, 280, 0, 1, 0, 0, 168, 0, 168, 0, 8, 0, 0, 420, 0, 56, 0, 56, 0, 0, 560, 0, 8, 0, 168, 56, 0, 420, 0, 0, 0, 280, 280, 0, 168

Offset: 8

Ilya Gutkovskiy, Feb 01 2021

Robert Israel, Table of n, a(n) for n = 8..10000
Index entries for sequences related to sums of cubes

Cf. A000578, A010057, A025461, A051344, A173681, A280618, A340977, A340978, A340979, A340980, 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, 8):
seq(a(n), n=8..95);  # Alois P. Heinz, Feb 01 2021

nmax = 95; CoefficientList[Series[Sum[x^(k^3), {k, 1, Floor[nmax^(1/3)] + 1}]^8, {x, 0, nmax}], x] // Drop[#, 8] &

G.f.: (Sum_{k>=1} x^(k^3))^8.

A340982 Number of ways to write n as an ordered sum of 9 positive cubes.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 36, 0, 0, 0, 0, 0, 0, 84, 0, 0, 0, 0, 9, 0, 126, 0, 0, 0, 0, 72, 0, 126, 0, 0, 0, 0, 252, 0, 84, 0, 0, 0, 0, 504, 0, 36, 0, 0, 36, 0, 630, 0, 9, 0, 0, 252, 0, 504, 0, 10, 0, 0, 756, 0, 252, 0, 72, 0, 0, 1260, 0, 72, 0, 252, 84, 0, 1260, 0, 9, 0, 504, 504

Offset: 9

Ilya Gutkovskiy, Feb 01 2021

nonn

Robert Israel, Table of n, a(n) for n = 9..10000
Index entries for sequences related to sums of cubes

Cf. A000578, A010057, A025462, A051344, A173682, A280618, A340977, A340978, A340979, A340980, A340981, 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, 9):
seq(a(n), n=9..94);  # Alois P. Heinz, Feb 01 2021

nmax = 94; CoefficientList[Series[Sum[x^(k^3), {k, 1, Floor[nmax^(1/3)] + 1}]^9, {x, 0, nmax}], x] // Drop[#, 9] &

G.f.: (Sum_{k>=1} x^(k^3))^9.

A340983 Number of ways to write n as an ordered sum of 10 positive cubes.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 45, 0, 0, 0, 0, 0, 0, 120, 0, 0, 0, 0, 10, 0, 210, 0, 0, 0, 0, 90, 0, 252, 0, 0, 0, 0, 360, 0, 210, 0, 0, 0, 0, 840, 0, 120, 0, 0, 45, 0, 1260, 0, 45, 0, 0, 360, 0, 1260, 0, 20, 0, 0, 1260, 0, 840, 0, 91, 0, 0, 2520, 0, 360, 0, 360, 120, 0, 3150, 0, 90

Offset: 10

Ilya Gutkovskiy, Feb 01 2021

Robert Israel, Table of n, a(n) for n = 10..10000
Index entries for sequences related to sums of cubes

Cf. A000578, A010057, A025463, A051344, A280618, A340977, A340978, A340979, A340980, A340981, A340982.

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, 10):
seq(a(n), n=10..92);  # Alois P. Heinz, Feb 01 2021

nmax = 92; CoefficientList[Series[Sum[x^(k^3), {k, 1, Floor[nmax^(1/3)] + 1}]^10, {x, 0, nmax}], x] // Drop[#, 10] &

G.f.: (Sum_{k>=1} x^(k^3))^10.

A347715 Number of compositions (ordered partitions) of n into at most 3 cubes.

Original entry on oeis.org

1, 1, 1, 1, 0, 0, 0, 0, 1, 2, 3, 0, 0, 0, 0, 0, 1, 3, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 2, 3, 0, 0, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 0, 0, 0, 0, 0, 0, 3, 0, 1, 2, 3, 0, 0, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 0, 3, 1

Offset: 0

Ilya Gutkovskiy, Sep 10 2021

nonn

Index entries for sequences related to sums of cubes

Cf. A000578, A023358, A051344, A347714, A347716.

A303485 a(n) = [x^(n^3)] (1/(1 - x))*(Sum_{k>=1} x^(k^3))^n.

Original entry on oeis.org

1, 1, 1, 8, 66, 512, 5269, 57459, 711742, 9610222, 139735699, 2183555015, 36543300668, 649320343729, 12174674648730, 240360451018461, 4975239937954534, 107600744797471150, 2426579187889852885, 56901290353169050995, 1384258146777832889697

Offset: 0

Ilya Gutkovskiy, Apr 24 2018

nonn

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

Index entries for sequences related to sums of cubes

Cf. A000578, A010057, A051344, A280618, A298672, A302995, A303169.

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

cp's OEIS Frontend

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

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A340977 Number of ways to write n as an ordered sum of 4 positive cubes.

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A340978 Number of ways to write n as an ordered sum of 5 positive cubes.

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

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A340980 Number of ways to write n as an ordered sum of 7 positive cubes.

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A340981 Number of ways to write n as an ordered sum of 8 positive cubes.

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A340982 Number of ways to write n as an ordered sum of 9 positive cubes.

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A340983 Number of ways to write n as an ordered sum of 10 positive cubes.

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