Displaying 1-10 of 11 results found.
Number of ways of writing n as a sum of 3 nonnegative cubes (counted naively).
+10
15
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
MAPLE
series(add(x^(n^3), n=0..10)^3, x, 1000);
PROG
(PARI) 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
CROSSREFS
Sums of k cubes, number of ways of writing n as, for k=1..9: A010057, A173677, A051343, A173678, A173679, A173680, A173676, A173681, A173682.
Number of ways of writing n as a sum of two nonnegative cubes.
+10
14
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
COMMENTS
Order matters. This is the coefficient of q^n in the expansion of {Sum_{m>=0} q^(m^3)}^2.
PROG
(PARI) 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
CROSSREFS
Sums of k cubes, number of ways of writing n as, for k=1..9: A010057, A173677, A051343, A173678, A173679, A173680, A173676, A173681, A173682.
Number of ways of writing n as a sum of 9 nonnegative cubes.
+10
13
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
COMMENTS
Order matters. This is the coefficient of q^n in the expansion of {Sum_{m>=0} q^(m^3)}^9.
CROSSREFS
Sums of k cubes, number of ways of writing n as, for k=1..9: A010057, A173677, A051343, A173678, A173679, A173680, A173676, A173681, A173682.
Number of ways of writing n as a sum of 4 nonnegative cubes.
+10
12
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
COMMENTS
Order matters. This is the coefficient of q^n in the expansion of {Sum_{m>=0} q^(m^3)}^4.
CROSSREFS
Sums of k cubes, number of ways of writing n as, for k=1..9: A010057, A173677, A051343, A173678, A173679, A173680, A173676, A173681, A173682.
Number of ways of writing n as a sum of 5 nonnegative cubes.
+10
12
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
COMMENTS
Order matters. This is the coefficient of q^n in the expansion of {Sum_{m>=0} q^(m^3)}^5.
CROSSREFS
Sums of k cubes, number of ways of writing n as, for k=1..9: A010057, A173677, A051343, A173678, A173679, A173680, A173676, A173681, A173682.
Number of ways of writing n as a sum of 6 nonnegative cubes.
+10
12
1, 6, 15, 20, 15, 6, 1, 0, 6, 30, 60, 60, 30, 6, 0, 0, 15, 60, 90, 60, 15, 0, 0, 0, 20, 60, 60, 26, 30, 60, 60, 30, 21, 30, 15, 30, 120, 180, 120, 30, 6, 6, 0, 60, 180, 180, 60, 0, 1, 0, 0, 60, 120, 60, 15, 60, 90, 60, 15, 30, 30, 0, 60, 180, 186, 90, 60, 66, 30, 6, 90, 180, 120, 120, 180, 120, 30, 0, 60, 60, 60, 200, 240, 120, 20, 0, 15, 0, 60, 180, 180
COMMENTS
Order matters. This is the coefficient of q^n in the expansion of {Sum_{m>=0} q^(m^3)}^6.
CROSSREFS
Sums of k cubes, number of ways of writing n as, for k=1..9: A010057, A173677, A051343, A173678, A173679, A173680, A173676, A173681, A173682.
Number of ways of writing n as a sum of 8 nonnegative cubes.
+10
12
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
COMMENTS
Order matters. This is the coefficient of q^n in the expansion of {Sum_{m>=0} q^(m^3)}^8.
PROG
(PARI) 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
CROSSREFS
Sums of k cubes, number of ways of writing n as, for k=1..9: A010057, A173677, A051343, A173678, A173679, A173680, A173676, A173681, A173682.
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.
+10
8
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
COMMENTS
A(n,k) is the number of ways of writing n as a sum of k nonnegative cubes.
FORMULA
G.f. of column k: (Sum_{j>=0} x^(j^3))^k.
EXAMPLE
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, ...
MATHEMATICA
Table[Function[k, SeriesCoefficient[Sum[x^i^3, {i, 0, n}]^k, {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten
Number of ways to write n as an ordered sum of 7 positive cubes.
+10
8
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
FORMULA
G.f.: (Sum_{k>=1} x^(k^3))^7.
MAPLE
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):
MATHEMATICA
nmax = 96; CoefficientList[Series[Sum[x^(k^3), {k, 1, Floor[nmax^(1/3)] + 1}]^7, {x, 0, nmax}], x] // Drop[#, 7] &
CROSSREFS
Cf. A000578, A010057, A025460, A051344, A173676, A280618, A340977, A340978, A340979, A340981, A340982, A340983.
Numbers that are the sum of at most 7 positive cubes.
+10
7
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
COMMENTS
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
CROSSREFS
Sums of k cubes, number of ways of writing n as, for k=1..9: A010057, A173677, A051343, A173678, A173679, A173680, A173676, A173681, A173682.
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