A023358 -id:A023358 - cp's OEIS frontend

A290247 Number of compositions (ordered partitions) of n^3 into cubes.

1, 1, 2, 120, 290250, 107320441096, 21715974961480054078, 8487986089807555456140271121440, 22615863021403796876556069287242400147213424924, 1449638083412288206280215383952017948209203861522683138464747658192

Offset: 0

Ilya Gutkovskiy, Jul 24 2017

nonn

			a(2) = 2 because 2^3 = 8 and we have [8], [1, 1, 1, 1, 1, 1, 1, 1].

Alois P. Heinz, Table of n, a(n) for n = 0..22
Index entries for sequences related to compositions

Cf. A000578, A023358, A030272, A224366, A259792.

b:= proc(n) option remember; local i; if n=0 then 1
      else 0; for i while i^3<=n do %+b(n-i^3) od fi
    end:
a:= n-> b(n^3):
seq(a(n), n=0..10);  # Alois P. Heinz, Aug 12 2017

Table[SeriesCoefficient[1/(1 - Sum[x^k^3, {k, 1, n}]), {x, 0, n^3}], {n, 0, 9}]

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

a(n) = A023358(A000578(n)).

A323633 Expansion of 1/Sum_{k>=0} x^(k^3).

Original entry on oeis.org

1, -1, 1, -1, 1, -1, 1, -1, 0, 1, -2, 3, -4, 5, -6, 7, -7, 6, -4, 1, 3, -8, 14, -21, 28, -34, 38, -40, 38, -31, 18, 2, -29, 62, -99, 139, -178, 211, -232, 234, -210, 154, -62, -70, 242, -449, 680, -917, 1135, -1303, 1386, -1344, 1136, -725, 85, 794, -1898, 3183, -4571

Offset: 0

Ilya Gutkovskiy, Jan 21 2019

sign

Convolution inverse of A010057.

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

Cf. A010057, A023358, A317665.

a:=series(1/add(x^(k^3),k=0..100),x=0,59): seq(coeff(a,x,n),n=0..58); # Paolo P. Lava, Apr 02 2019

nmax = 58; CoefficientList[Series[1/Sum[x^k^3, {k, 0, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = -Sum[Boole[IntegerQ[k^(1/3)]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 58}]

my(N=66, x='x+O('x^N)); Vec(1/sum(k=0, N^(1/3), x^k^3)) \\ Seiichi Manyama, Mar 19 2022

a(n) = if(n==0, 1, -sum(k=1, n, ispower(k, 3)*a(n-k))); \\ Seiichi Manyama, Mar 19 2022

a(0) = 1; a(n) = -Sum_{k=1..n} A010057(k) * a(n-k). - Seiichi Manyama, Mar 19 2022

A331845 Number of compositions (ordered partitions) of n into distinct cubes.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 2, 6, 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, 1, 2, 0, 0, 0, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 0, 6, 24

Offset: 0

Ilya Gutkovskiy, Jan 29 2020

			a(36) = 6 because we have [27,8,1], [27,1,8], [8,27,1], [8,1,27], [1,27,8] and [1,8,27].

Cf. A000578, A023358, A032020, A032021, A032022, A218396, A219107, A279329, A331843, A331844, A331846, A331847.

b:= proc(n, i, p) option remember;
      `if`((i*(i+1)/2)^2n, 0, b(n-i^3, i-1, p+1))+b(n, i-1, p)))
    end:
a:= n-> b(n, iroot(n, 3), 0):
seq(a(n), n=0..100);  # Alois P. Heinz, Jan 30 2020

b[n_, i_, p_] := b[n, i, p] = If[(i(i+1)/2)^2 < n, 0, If[n == 0, p!, If[i^3 > n, 0, b[n-i^3, i-1, p+1]] + b[n, i-1, p]]];
a[n_] := b[n, Floor[n^(1/3)], 0];
a /@ Range[0, 100] (* Jean-François Alcover, Oct 31 2020, after Alois P. Heinz *)

A347714 Number of compositions (ordered partitions) of n into at most 2 cubes.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Sep 10 2021

nonn

Index entries for sequences related to sums of cubes

Cf. A000578, A023358, A280618, A347715, A347716.

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.

A347716 Number of compositions (ordered partitions) of n into at most 4 cubes.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 0, 0, 1, 2, 3, 4, 0, 0, 0, 0, 1, 3, 6, 0, 0, 0, 0, 0, 1, 4, 0, 1, 2, 3, 4, 0, 1, 0, 0, 2, 6, 12, 0, 0, 0, 0, 0, 3, 12, 0, 0, 0, 0, 0, 0, 4, 0, 0, 1, 3, 6, 0, 0, 0, 0, 0, 3, 12, 1, 2, 3, 4, 0, 0, 6, 0, 2, 6, 12, 0, 0, 0, 0, 0, 3, 13, 4, 0, 0

Offset: 0

Ilya Gutkovskiy, Sep 10 2021

nonn

Index entries for sequences related to sums of cubes

Cf. A000578, A023358, A340977, A347714, A347715.

A246885 Those n for which the coefficients of x^n in the reciprocal of 1+x+x^8+...+x^(i^3)+... are odd.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 11, 13, 15, 16, 19, 20, 23, 29, 32, 34, 35, 37, 45, 47, 48, 49, 53, 54, 57, 58, 67, 69, 71, 73, 75, 85, 86, 99, 101, 107, 108, 109, 110, 115, 121, 123, 124, 127, 128, 129, 131, 132, 135, 137, 141, 143, 155, 157, 160, 161, 162, 163, 169, 177, 183, 189, 193, 195, 197, 199, 203

Offset: 1

David S. Newman, Sep 06 2014

nonn

Numbers n such that the number of compositions of n into cubes (A023358) is odd. - Joerg Arndt, Sep 08 2014

			The reciprocal of 1+x+x^8+x^27+... begins 1 -x +x^2 -x^3 +x^4 -x^5 +x^6 -x^7 +x^9 -2*x^10 +...  So the first few values of a(n) are 0,1,2,3,4,5,6,7,9... .

Alois P. Heinz, Table of n, a(n) for n = 1..10000
J. N. Cooper, D. Eichhorn and K. O'Bryant, Reciprocals of binary power series, arXiv:math/0506496 [math.NT], 2005.

Cf. A023358.

b:= proc(n) option remember; irem(`if`(n=0, 1,
      `if`(n<0, 0, add(b(n-i^3), i=1..iroot(n, 3)))), 2)
    end:
a:= proc(n) option remember; local k; for k from 1+
      `if`(n=1, -1, a(n-1)) while b(k)=0 do od; k
    end:
seq(a(n), n=1..80);  # Alois P. Heinz, Sep 08 2014

iend=10;
seq=Flatten[Position[Delete[Mod[CoefficientList[Series[1/Sum[x^(i^3),{i,0,iend}],{x,0,iend^3}],x],2],1],1]];
Print[seq];

A278929 Expansion of 1/(1 - Sum_{k>=1} x^(prime(k)^3)).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 3, 0, 0, 0, 0, 1, 0, 0, 4, 0, 0, 1, 0, 1, 0, 0, 5, 0, 0, 3, 0, 1, 0, 0, 6, 0, 0, 6, 0, 1, 0, 0, 7, 0, 0, 10, 0, 1, 1, 0, 8, 0, 0, 15, 0, 1, 4, 0, 9, 0, 0, 21

Offset: 0

Ilya Gutkovskiy, Dec 24 2016

nonn

Number of compositions (ordered partitions) of n into cubes of primes (A030078).

			a(35) = 2 because we have [8, 27] and [27, 8].

Robert Israel, Table of n, a(n) for n = 0..10000
Index entries for sequences related to sums of cubes

Cf. A023358, A023360, A030078, A279760.

N:= 200:
Primes:= select(isprime, [2,seq(i,i=3..floor(N^(1/3)),2)]):
G:= 1/(1- add(x^(Primes[i]^3),i=1..nops(Primes))):
S:= series(G,x,N+1):
seq(coeff(S,x,j),j=0..N); # Robert Israel, Jan 23 2019

nmax = 120; CoefficientList[Series[1/(1 - Sum[x^Prime[k]^3, {k, 1, nmax}]), {x, 0, nmax}], x]

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

A298936 Number of ordered ways of writing n^2 as a sum of n nonnegative cubes.

Original entry on oeis.org

1, 1, 0, 6, 6, 20, 120, 7, 1689, 6636, 36540, 64020, 963996, 2894892, 19555965, 176079995, 955611188, 6684303780, 42462792168, 292378003753, 1886275214112, 13384059605364, 87399249887334, 624073002367892, 5080120229014734, 37587589611771480

Offset: 0

Ilya Gutkovskiy, Jan 29 2018

nonn

			a(3) = 6 because we have [8, 1, 0], [8, 0, 1], [1, 8, 0], [1, 0, 8], [0, 8, 1] and [0, 1, 8].

Index entries for sequences related to sums of cubes

Cf. A000290, A000578, A023358, A030272, A218495, A259792, A291700, A298671, A298672, A298934, A298937.

f:= n -> coeff(add(x^(k^3),k=0..floor(n^(2/3)))^n,x,n^2):
map(f, [$0..30]); # Robert Israel, Jan 29 2018

Table[SeriesCoefficient[Sum[x^k^3, {k, 0, Floor[n^(2/3) + 1]}]^n, {x, 0, n^2}], {n, 0, 25}]

a(n) = [x^(n^2)] (Sum_{k>=0} x^(k^3))^n.

A298937 Number of ordered ways of writing n^2 as a sum of n positive cubes.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 7, 1, 0, 0, 9240, 34650, 1716, 48477, 551915, 6726720, 89973520, 102639744, 1824625081, 9915389400, 30143458884, 278196062760, 1995766236541, 6611689457736, 64547920386450, 236756174748626, 2315743488707806

Offset: 0

Ilya Gutkovskiy, Jan 29 2018

nonn

			a(7) = 7 because we have [8, 8, 8, 8, 8, 8, 1], [8, 8, 8, 8, 8, 1, 8], [8, 8, 8, 8, 1, 8, 8], [8, 8, 8, 1, 8, 8, 8], [8, 8, 1, 8, 8, 8, 8], [8, 1, 8, 8, 8, 8, 8] and [1, 8, 8, 8, 8, 8, 8].

Index entries for sequences related to sums of cubes

Cf. A000290, A000578, A023358, A030272, A218495, A259792, A291700, A298671, A298672, A298934, A298936.

Join[{1}, Table[SeriesCoefficient[Sum[x^k^3, {k, 1, Floor[n^(2/3) + 1]}]^n, {x, 0, n^2}], {n, 1, 27}]]

a(n) = [x^(n^2)] (Sum_{k>=1} x^(k^3))^n.

cp's OEIS Frontend

A290247 Number of compositions (ordered partitions) of n^3 into cubes.

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A323633 Expansion of 1/Sum_{k>=0} x^(k^3).

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A331845 Number of compositions (ordered partitions) of n into distinct cubes.

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A347714 Number of compositions (ordered partitions) of n into at most 2 cubes.

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

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A347716 Number of compositions (ordered partitions) of n into at most 4 cubes.

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A246885 Those n for which the coefficients of x^n in the reciprocal of 1+x+x^8+...+x^(i^3)+... are odd.

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A278929 Expansion of 1/(1 - Sum_{k>=1} x^(prime(k)^3)).

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A298936 Number of ordered ways of writing n^2 as a sum of n nonnegative cubes.

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