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.

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A341791 Number of partitions of n into 8 nonzero tetrahedral numbers.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 2, 0, 0, 3, 1, 0, 3, 1, 0, 3, 1, 0, 3, 2, 0, 3, 2, 0, 2, 3, 0, 3, 4, 1, 2, 4, 1, 2, 4, 1, 2, 5, 2, 2, 4, 2, 1, 5, 3, 2, 6, 4, 2, 5, 3, 2, 5, 4, 2, 6, 4, 3, 5, 5, 2, 5, 5, 4, 6, 6, 3, 6, 6, 3, 5, 6, 3, 6, 8, 4, 5, 8, 4, 5, 8, 4, 5, 10
Offset: 8

Views

Author

Ilya Gutkovskiy, Feb 19 2021

Keywords

Crossrefs

Cf. A000292, A023533, A024692, A068980, A319818, A341774, A341775, A341776, A341777, A341778, A341792, A341793.

A341792 Number of partitions of n into 9 nonzero tetrahedral numbers.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 2, 0, 0, 3, 1, 0, 3, 1, 0, 3, 1, 0, 4, 2, 0, 3, 2, 0, 3, 3, 0, 3, 4, 1, 3, 4, 1, 2, 5, 1, 3, 5, 2, 2, 5, 2, 2, 5, 3, 2, 7, 4, 3, 6, 4, 2, 6, 4, 3, 7, 5, 3, 6, 5, 3, 6, 6, 4, 7, 7, 5, 7, 7, 3, 7, 7, 5, 7, 9, 4, 7, 9, 5, 6, 10, 5, 8
Offset: 9

Views

Author

Ilya Gutkovskiy, Feb 19 2021

Keywords

Crossrefs

Cf. A000292, A023533, A024692, A068980, A319819, A341774, A341775, A341776, A341777, A341778, A341791, A341793.

A341793 Number of partitions of n into 10 nonzero tetrahedral numbers.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 2, 0, 0, 3, 1, 0, 3, 1, 0, 3, 1, 0, 4, 2, 0, 4, 2, 0, 3, 3, 0, 4, 4, 1, 3, 4, 1, 3, 5, 1, 3, 6, 2, 3, 5, 2, 2, 6, 3, 3, 7, 4, 3, 7, 4, 3, 7, 5, 3, 8, 5, 4, 7, 6, 3, 7, 6, 5, 8, 8, 5, 8, 8, 5, 8, 8, 5, 9, 10, 6, 8, 10, 5, 8, 11, 7
Offset: 10

Views

Author

Ilya Gutkovskiy, Feb 19 2021

Keywords

Crossrefs

Cf. A000292, A023533, A024692, A068980, A319820, A341774, A341775, A341776, A341777, A341778, A341791, A341792.

A279757 Expansion of Product_{k>=1} 1/(1 - x^(k*(2*k^2+1)/3)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 5, 5, 5, 5, 5, 6, 7, 7, 7, 7, 7, 8, 9, 9, 9, 9, 9, 10, 11, 12, 12, 12, 12, 13, 14, 16, 16, 16, 16, 17, 18, 20, 20, 20, 20, 21, 22, 24, 25, 25, 25, 26, 27, 29, 31, 31, 31, 32, 33, 35, 37, 37, 37, 38, 39, 41, 43, 44, 44, 45, 46, 48, 50, 52, 52, 53, 55, 57, 59, 62, 62, 63, 65, 67, 69, 72, 73
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 18 2016

Keywords

Comments

Number of partitions of n into nonzero octahedral numbers (A005900).

Examples

			a(7) = 2 because we have [6, 1] and [1, 1, 1, 1, 1, 1, 1].

Links

Crossrefs

Cf. A003108, A005900, A068980, A279758, A279759.

Programs

  • Mathematica
    nmax=95; CoefficientList[Series[Product[1/(1 - x^(k (2 k^2 + 1)/3)), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

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

A279758 Expansion of Product_{k>=1} 1/(1 - x^(k*(5*k^2-5*k+2)/2)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 18 2016

Keywords

Comments

Number of partitions of n into nonzero icosahedral numbers (A006564).

Examples

			a(13) = 2 because we have [12, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1].

Links

Crossrefs

Cf. A003108, A006564, A068980, A279757, A279759.

Programs

  • Mathematica
    nmax=105; CoefficientList[Series[Product[1/(1 - x^(k (5 k^2 - 5 k + 2)/2)), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} 1/(1 - x^(k*(5*k^2-5*k+2)/2)).

A279759 Expansion of Product_{k>=1} 1/(1 - x^(k*(3*k-1)*(3*k-2)/2)).

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, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 18 2016

Keywords

Comments

Number of partitions of n into nonzero dodecahedral numbers (A006566).

Examples

			a(21) = 2 because we have [20, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1].

Links

Crossrefs

Cf. A003108, A006566, A068980, A279757, A279758.

Programs

  • Mathematica
    nmax=120; CoefficientList[Series[Product[1/(1 - x^(k (3 k - 1) (3 k - 2)/2)), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

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

A290573 Expansion of Product_{k>=1} 1/(1 - x^(k*(k+1)*(k+2)*(k+3)/24)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 9, 9, 9, 9, 9, 12, 12, 12, 12, 12, 16, 16, 16, 16, 16, 20, 20, 20, 20, 20, 25, 25, 25, 25, 25, 31, 31, 31, 31, 31, 37, 37, 37, 37, 37, 44, 44, 44, 44, 44, 52, 52, 52, 52, 52, 62, 62, 62, 62, 62, 73, 73, 73, 73, 73, 85, 85, 85, 85, 85, 99
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 15 2017

Keywords

Comments

Number of partitions of n into nonzero 4-dimensional pyramidal numbers (A000332).

Examples

			a(10) = 3 because we have [5, 5], [5, 1, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1].

Links

Crossrefs

Cf. A000292, A000332, A007294, A068980, A290792.

Programs

  • Mathematica
    nmax = 85; CoefficientList[Series[Product[1/(1 - x^(k (k + 1) (k + 2) (k + 3)/24)), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} 1/(1 - x^(k*(k+1)*(k+2)*(k+3)/24)).

A298247 Expansion of Product_{k>=1} (1 - x^(k*(k+1)*(k+2)/6)).

Original entry on oeis.org

1, -1, 0, 0, -1, 1, 0, 0, 0, 0, -1, 1, 0, 0, 1, -1, 0, 0, 0, 0, -1, 1, 0, 0, 1, -1, 0, 0, 0, 0, 1, -1, 0, 0, -1, 0, 1, 0, 0, 1, -1, 0, 0, 0, 0, 1, -1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -2, 1, 0, -1, 2, -1, 0, 0, 0, -1, 2, -1, 0, 1, -2, 1, 0, 0, 0, 0, 1, -1, 0, 0, -1, 1, 0, 0, -1, 1, -1, 1, 1, -1, 1, 0, -1, 0, 1, -2, 1, 0, -1, 1, 0, -1, 1, 0, 1
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 15 2018

Keywords

Comments

The difference between the number of partitions of n into an even number of distinct tetrahedral numbers and the number of partitions of n into an odd number of distinct tetrahedral numbers.

Links

Crossrefs

Cf. A000292, A068980, A279278, A292518, A298249.

Programs

  • Mathematica
    nmax = 104; CoefficientList[Series[Product[1 - x^(k (k + 1) (k + 2)/6), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} (1 - x^A000292(k)).

A334988 Sum of tetrahedral numbers dividing n.

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 1, 5, 1, 11, 1, 5, 1, 1, 1, 5, 1, 1, 1, 35, 1, 1, 1, 5, 1, 1, 1, 5, 1, 11, 1, 5, 1, 1, 36, 5, 1, 1, 1, 35, 1, 1, 1, 5, 1, 1, 1, 5, 1, 11, 1, 5, 1, 1, 1, 61, 1, 1, 1, 35, 1, 1, 1, 5, 1, 1, 1, 5, 1, 46, 1, 5, 1, 1, 1, 5, 1, 1, 1, 35, 1, 1, 1, 89, 1, 1, 1, 5, 1, 11
Offset: 1

Views

Author

Ilya Gutkovskiy, May 18 2020

Keywords

Links

Crossrefs

Cf. A000292, A023533, A068980, A185027, A279495, A334987.

Programs

  • Mathematica
    nmax = 90; CoefficientList[Series[Sum[Binomial[k + 2, 3] x^Binomial[k + 2, 3]/(1 - x^Binomial[k + 2, 3]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
nmax = 90; CoefficientList[Series[Log[Product[1/(1 - x^Binomial[k + 2, 3]), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax] // Rest
  • PARI
    ist(n) = my(k=sqrtnint(6*n, 3)); k*(k+1)*(k+2)==6*n; \\ A000292
a(n) = sumdiv(n, d, if (ist(d), d)); \\ Michel Marcus, May 19 2020

Formula

G.f.: Sum_{k>=1} binomial(k+2,3) * x^binomial(k+2,3) / (1 - x^binomial(k+2,3)).
L.g.f.: log(G(x)), where G(x) is the g.f. for A068980.
a(n) = Sum_{d|n} A023533(d) * d.

A341773 Number of partitions of 2*n into exactly n nonzero tetrahedral numbers.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 2, 0, 0, 3, 1, 0, 3, 1, 0, 3, 1, 0, 4, 2, 0, 4, 2, 0, 4, 3, 0, 5, 4, 1, 5, 4, 1, 5, 5, 1, 6, 6, 2, 6, 6, 2, 6, 7, 3, 7, 9, 4, 8, 9, 4, 8, 10, 5, 9, 12, 6, 10, 12, 7, 10, 13, 8, 12, 15, 10, 13, 16, 11, 13, 17, 12
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 19 2021

Keywords

Links

Crossrefs

Cf. A000292, A024692, A062748, A068980, A298269, A319799, A338586, A341774, A341775, A341776, A341777, A341778.

Programs

  • Mathematica
    nmax = 80; CoefficientList[Series[Product[1/(1 - x^(Binomial[k + 4, 3] - 1)), {k, 0, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=0} 1 / (1 - x^(binomial(k+4,3)-1)).
Previous Showing 21-30 of 31 results. Next

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