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.

Showing 1-2 of 2 results.

A280718 Expansion of (Sum_{k>=0} x^(k*(3*k-1)/2))^5.

Original entry on oeis.org

1, 5, 10, 10, 5, 6, 20, 30, 20, 5, 10, 30, 35, 30, 30, 30, 25, 30, 60, 60, 25, 5, 35, 80, 70, 51, 35, 50, 80, 90, 80, 30, 35, 60, 80, 95, 90, 90, 50, 75, 140, 140, 85, 20, 70, 120, 130, 120, 95, 115, 100, 115, 140, 155, 110, 40, 80, 200, 230, 140, 81, 120, 200, 190, 180, 120, 80, 100, 160, 240, 200, 155, 120, 140, 245, 260, 230
Offset: 0

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Author

Ilya Gutkovskiy, Feb 10 2017

Keywords

Comments

Number of ways to write n as an ordered sum of 5 pentagonal numbers (A000326).
a(n) > 0 for all n >= 0.
Every number is the sum of at most 5 pentagonal numbers.
Every number is the sum of at most k k-gonal numbers (Fermat's polygonal number theorem).

Examples

			a(5) = 6 because we have:
[5, 0, 0, 0, 0]
[0, 5, 0, 0, 0]
[0, 0, 5, 0, 0]
[0, 0, 0, 5, 0]
[0, 0, 0, 0, 5]
[1, 1, 1, 1, 1]

Links

Crossrefs

Cf. A000326, A003679, A008439, A038671, A280719, A282248.

Programs

  • Mathematica
    nmax = 76; CoefficientList[Series[Sum[x^(k (3 k - 1)/2), {k, 0, nmax}]^5, {x, 0, nmax}], x]

Formula

G.f.: (Sum_{k>=0} x^(k*(3*k-1)/2))^5.

A280719 Expansion of (Sum_{k>=0} x^(k*(2*k-1)))^6.

Original entry on oeis.org

1, 6, 15, 20, 15, 6, 7, 30, 60, 60, 30, 6, 15, 60, 90, 66, 45, 60, 80, 90, 66, 50, 120, 180, 135, 60, 15, 60, 186, 210, 141, 126, 120, 126, 165, 180, 241, 300, 210, 90, 90, 180, 270, 270, 210, 212, 270, 270, 200, 210, 366, 450, 390, 270, 135, 210, 375, 360, 396, 420, 300, 330, 375, 380, 510, 480, 336, 450, 510, 390, 330
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 10 2017

Keywords

Comments

Number of ways to write n as an ordered sum of 6 hexagonal numbers (A000384).
a(n) > 0 for all n >= 0.
Every number is the sum of at most 6 hexagonal numbers.
Every number is the sum of at most k k-gonal numbers (Fermat's polygonal number theorem).

Examples

			a(6) = 7 because we have:
[6, 0, 0, 0, 0, 0]
[0, 6, 0, 0, 0, 0]
[0, 0, 6, 0, 0, 0]
[0, 0, 0, 6, 0, 0]
[0, 0, 0, 0, 6, 0]
[0, 0, 0, 0, 0, 6]
[1, 1, 1, 1, 1, 1]

Links

Crossrefs

Cf. A000384, A007536, A008440, A045848, A280718, A282248.

Programs

  • Mathematica
    nmax = 70; CoefficientList[Series[Sum[x^(k (2 k - 1)), {k, 0, nmax}]^6, {x, 0, nmax}], x]

Formula

G.f.: (Sum_{k>=0} x^(k*(2*k-1)))^6.
Showing 1-2 of 2 results.

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