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.

Previous Showing 11-20 of 23 results. Next

A339420 Number of compositions (ordered partitions) of n into an even number of cubes.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 0, 1, 2, 1, 4, 1, 6, 1, 8, 2, 10, 7, 12, 16, 14, 29, 16, 46, 22, 67, 40, 94, 78, 125, 144, 161, 246, 214, 394, 312, 602, 499, 878, 835, 1236, 1396, 1722, 2286, 2446, 3637, 3614, 5598, 5560, 8358, 8782, 12226, 14014, 17776, 22278, 26056, 34924
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 03 2020

Keywords

Examples

			a(11) = 4 because we have [8, 1, 1, 1], [1, 8, 1, 1], [1, 1, 8, 1] and [1, 1, 1, 8].

Links

Crossrefs

Cf. A000578, A023358, A034008, A323633, A339368, A339418, A339421.

Programs

  • Maple
    b:= proc(n, t) option remember; local r, f, g;
      if n=0 then t else r, f, g:=$0..2; while f<=n
      do r, f, g:= r+b(n-f, 1-t), f+3*g*(g-1)+1, g+1 od; r fi
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..60);  # Alois P. Heinz, Dec 03 2020
  • Mathematica
    nmax = 57; CoefficientList[Series[(1/2) (1/(1 - Sum[x^(k^3), {k, 1, Floor[nmax^(1/3)] + 1}]) + 1/Sum[x^(k^3), {k, 0, Floor[nmax^(1/3)] + 1}]), {x, 0, nmax}], x]

Formula

G.f.: (1/2) * (1 / (1 - Sum_{k>=1} x^(k^3)) + 1 / Sum_{k>=0} x^(k^3)).
a(n) = (A023358(n) + A323633(n)) / 2.
a(n) = Sum_{k=0..n} A023358(k) * A323633(n-k).

A339421 Number of compositions (ordered partitions) of n into an odd number of cubes.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 3, 1, 5, 1, 7, 1, 9, 4, 11, 11, 13, 22, 15, 37, 18, 56, 29, 80, 56, 109, 107, 142, 190, 184, 313, 255, 490, 391, 731, 644, 1045, 1082, 1458, 1792, 2044, 2895, 2957, 4531, 4463, 6863, 6972, 10126, 11090, 14739, 17691, 21484, 27954, 31741
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 03 2020

Keywords

Examples

			a(10) = 3 because we have [8, 1, 1], [1, 8, 1] and [1, 1, 8].

Links

Crossrefs

Cf. A000578, A023358, A166444, A323633, A339369, A339419, A339420.

Programs

  • Maple
    b:= proc(n, t) option remember; local r, f, g;
      if n=0 then t else r, f, g:=$0..2; while f<=n
      do r, f, g:= r+b(n-f, 1-t), f+3*g*(g-1)+1, g+1 od; r fi
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Dec 03 2020
  • Mathematica
    nmax = 57; CoefficientList[Series[(1/2) (1/(1 - Sum[x^(k^3), {k, 1, Floor[nmax^(1/3)] + 1}]) - 1/Sum[x^(k^3), {k, 0, Floor[nmax^(1/3)] + 1}]), {x, 0, nmax}], x]

Formula

G.f.: (1/2) * (1 / (1 - Sum_{k>=1} x^(k^3)) - 1 / Sum_{k>=0} x^(k^3)).
a(n) = (A023358(n) - A323633(n)) / 2.
a(n) = -Sum_{k=0..n-1} A023358(k) * A323633(n-k).

A347591 Number of compositions (ordered partitions) of n^3 into at most n cubes.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 27, 553, 3192, 87185, 999959, 10684689, 137722770, 2005577212, 27957554982, 492643033682, 8952039793647, 154671244623527, 3207929433418044, 66983196041550714, 1392059664888123313, 32337888832381327369, 763357156272340549200
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 08 2021

Keywords

Links

Crossrefs

Cf. A000578, A023358, A290247, A298672, A307738, A347590, A347592.

A348524 Number of compositions (ordered partitions) of n into two or more cubes.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 1, 1, 3, 4, 5, 6, 7, 8, 9, 11, 14, 18, 23, 29, 36, 44, 53, 64, 78, 96, 119, 150, 187, 232, 286, 351, 430, 527, 649, 802, 993, 1230, 1522, 1880, 2318, 2854, 3514, 4330, 5341, 6594, 8145, 10061, 12423, 15330, 18908, 23316, 28753, 35467, 43762
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 21 2021

Keywords

Links

Crossrefs

Cf. A000578, A010057, A023358, A347805, A348522.

Programs

  • Maple
    g:= proc(n) option remember;
     local i,m,t;
     m:= surd(n,3);
     if m::integer then t:= 1; m:= m-1 else t:= 0; m:= floor(m) fi;
     t + add(procname(n-i^3),i=1..m)
end proc:
f:= proc(n) local m;
    m:= surd(n,3);
    if m::integer then g(n)-1 else g(n) fi
end proc:
f(0):= 0:
map(f, [$0..100]);
  • Mathematica
    g[n_] := g[n] = Module[{m, t}, m = n^(1/3); If[IntegerQ[m], t = 1; m = m - 1, t = 0; m = Floor[m]]; t + Sum[g[n - i^3], {i, 1, m}]];
f[n_] := Module[{m}, m = n^(1/3); If[IntegerQ[m], g[n]-1, g[n]]];
f[0] = 0;
Map[f, Range[0, 100]] (* Jean-François Alcover, Sep 19 2022, after Robert Israel *)

A363748 Number of compositions into sums of fourth powers.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 22, 26, 31, 37, 44, 52, 61, 71, 82, 94, 107, 121, 136, 152, 169, 188, 210, 236, 267, 304, 348, 400, 461, 532, 614, 708, 815, 936, 1072, 1224, 1393, 1581, 1791, 2027, 2294, 2598, 2946, 3346, 3807, 4339, 4953, 5661, 6476, 7412, 8484, 9708, 11101, 12682, 14474
Offset: 0

Views

Author

Seiichi Manyama, Jun 19 2023

Keywords

Comments

This sequence is different from A291149.

Examples

			a(18)=4 counts the compositions 1^4+1^4+1^4+2^4 = 1^4+1^4+2^4+1^4 = 1^4+2^4+1^4+1^4 = 2^4+1^4+1^4+1^4. - _R. J. Mathar_, Jun 21 2023

Links

Crossrefs

Cf. A006456, A023358, A363749.
Cf. A000583, A291149, A352529.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, ispower(j, 4)*v[i-j+1])); v;

Formula

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

A363749 Number of compositions into sums of fifth powers.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 35, 38, 42, 47, 53, 60, 68, 77, 87, 98, 110, 123, 137, 152, 168, 185
Offset: 0

Views

Author

Seiichi Manyama, Jun 19 2023

Keywords

Comments

This sequence is different from A291168.

Links

Crossrefs

Cf. A006456, A023358, A363748.
Cf. A000584, A291168, A352530.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, ispower(j, 5)*v[i-j+1])); v;

Formula

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

A280865 Expansion of 1/(1 - Sum_{k>=0} x^((2*k+1)^3)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 30, 33, 37, 42, 48, 55, 63, 72, 82, 93, 105, 118, 132, 147, 163, 180, 198, 217, 237, 258, 280, 303, 327, 352, 378, 405, 433, 463, 496
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 09 2017

Keywords

Comments

Number of compositions (ordered partitions) of n into odd cubes (A016755).

Examples

			a(28) = 3 because we have [27, 1], [1, 27] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1].

Links

Crossrefs

Cf. A000578, A016755, A023358, A003108, A078128, A279329, A280130.

Programs

  • Mathematica
    nmax = 82; CoefficientList[Series[1/(1 - Sum[x^(2 k + 1)^3, {k, 0, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - Sum_{k>=0} x^((2*k+1)^3)).

A281809 Expansion of Sum_{i>=1} x^(i^3) / (1 - Sum_{j>=1} x^(j^3))^2.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 13, 19, 27, 37, 49, 63, 79, 99, 126, 163, 213, 279, 364, 471, 603, 766, 970, 1229, 1562, 1992, 2545, 3251, 4144, 5266, 6672, 8435, 10655, 13462, 17019, 21527, 27230, 34425, 43478, 54846, 69114, 87032, 109555, 137889, 173543, 218393, 274765, 345544, 434332, 545650, 685187, 860105, 1079402
Offset: 1

Views

Author

Ilya Gutkovskiy, Jan 30 2017

Keywords

Comments

Total number of parts in all compositions (ordered partitions) of n into cubes (A000578).

Examples

			a(10) = 19 because we have [8, 1, 1], [1, 8, 1], [1, 1, 8], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1] and 3 + 3 + 3 + 10 = 19.

Links

Crossrefs

Cf. A000578, A023358.

Programs

  • Maple
    b:= proc(n) option remember; `if`(n=0, [1, 0], add(
      (p-> p+[0, p[1]])(b(n-j^3)), j=1..iroot(n, 3)))
    end:
a:= n-> b(n)[2]:
seq(a(n), n=1..55);  # Alois P. Heinz, Aug 07 2019
  • Mathematica
    nmax = 55; Rest[CoefficientList[Series[Sum[x^i^3, {i, 1, nmax}]/(1 - Sum[x^j^3, {j, 1, nmax}])^2, {x, 0, nmax}], x]]

Formula

G.f.: Sum_{i>=1} x^(i^3) / (1 - Sum_{j>=1} x^(j^3))^2.

A347728 Number of compositions (ordered partitions) of n into at most 5 cubes.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 0, 0, 1, 2, 3, 4, 5, 0, 0, 0, 1, 3, 6, 10, 0, 0, 0, 0, 1, 4, 10, 1, 2, 3, 4, 5, 1, 5, 0, 2, 6, 12, 20, 0, 1, 0, 0, 3, 12, 30, 0, 0, 0, 0, 0, 4, 20, 0, 1, 3, 6, 10, 0, 5, 0, 0, 3, 12, 31, 2, 3, 4, 5, 0, 6, 30, 2, 6, 12, 20, 0, 0, 10, 0, 3, 13
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 11 2021

Keywords

Links

Crossrefs

Cf. A000578, A023358, A340978, A347714, A347715, A347716, A347729.

A347729 Number of compositions (ordered partitions) of n into at most 6 cubes.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 3, 4, 5, 6, 0, 0, 1, 3, 6, 10, 15, 0, 0, 0, 1, 4, 10, 21, 2, 3, 4, 5, 7, 5, 15, 2, 6, 12, 20, 30, 1, 6, 0, 3, 12, 30, 60, 0, 1, 0, 0, 4, 20, 60, 1, 3, 6, 10, 15, 5, 30, 0, 3, 12, 31, 62, 3, 10, 5, 6, 6, 30, 92, 6, 12, 20, 30, 0, 10, 60, 3, 13
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 11 2021

Keywords

Links

Crossrefs

Cf. A000578, A023358, A340979, A347714, A347715, A347716, A347728.
Previous Showing 11-20 of 23 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.