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 1-3 of 3 results.

A239000 Number of partitions of n using Fibonacci numbers > 2.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 0, 2, 1, 1, 2, 1, 3, 2, 2, 4, 2, 4, 4, 3, 7, 4, 5, 8, 5, 9, 8, 7, 12, 9, 11, 13, 11, 17, 14, 15, 20, 16, 22, 22, 20, 29, 24, 27, 33, 28, 37, 36, 35, 45, 40, 46, 50, 47, 60, 55, 58, 69, 62, 75, 76, 73, 91, 84, 91, 102, 95, 114, 112, 113
Offset: 0

Views

Author

Clark Kimberling, Mar 08 2014

Keywords

Examples

			a(21) counts these partitions: [21], [13,8], [13,5,3], [8,8,5], [8,5,5,3], [5,5,5,3,3], [3,3,3,3,3,3,3].

Links

Crossrefs

Cf. A000045, A003107, A238999.

Programs

  • Mathematica
    p[n_] := IntegerPartitions[n, All, Fibonacci@Range[4, 60]]; Table[p[n], {n, 0, 12}] (*shows partitions*)
a[n_] := Length@p@n; a /@ Range[0, 80] (*counts partitions, A239000*)
  • PARI
    N=66; q='q+O('q^N); Vec( 1/prod(n=1,11,1-q^fibonacci(n+3)) ) \\ Joerg Arndt, Mar 11 2014

Formula

G.f.: 1/Product_{i>=4} (1 - x^Fibonacci(i)).

A280168 Expansion of Product_{k>=2} 1/(1 - x^(Fibonacci(k)^2)).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 5, 6, 6, 6, 7, 8, 9, 9, 10, 11, 12, 12, 13, 15, 16, 17, 18, 20, 21, 22, 23, 25, 27, 28, 30, 32, 34, 35, 37, 39, 41, 43, 45, 48, 50, 52, 54, 57, 60, 62, 65, 68, 72, 74, 77, 80, 84, 87, 90, 94, 98, 102, 106, 110, 114, 118, 123, 127, 132, 136, 142, 147, 152, 157, 163, 169, 174, 180, 186, 193, 199
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 27 2016

Keywords

Comments

Number of partitions of n into squares of Fibonacci numbers (with a single type of 1).

Examples

			a(8) = 3 because we have [4, 4], [4, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1].

Links

Crossrefs

Cf. A000119, A000121, A001156, A003107, A007000, A007598, A238999, A239002.

Programs

  • Mathematica
    CoefficientList[Series[Product[1/(1 - x^Fibonacci[k]^2), {k, 2, 20}], {x, 0, 82}], x]

Formula

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

A303903 Expansion of (1 - x^2)*Product_{k>=3} (1 + x^Fibonacci(k)).

Original entry on oeis.org

1, 0, 0, 1, -1, 1, 0, -1, 2, -1, 0, 1, -2, 2, 0, -1, 2, -2, 1, 0, -2, 3, -1, 0, 2, -3, 2, 0, -2, 3, -2, 0, 1, -3, 3, 0, -1, 3, -3, 2, 0, -3, 4, -2, 0, 2, -4, 3, 0, -2, 3, -3, 1, 0, -3, 4, -1, 0, 3, -4, 3, 0, -3, 5, -3, 0, 2, -5, 4, 0, -2, 4, -4, 2, 0, -4, 5, -2, 0, 3, -5, 3, 0, -3, 4, -3
Offset: 0

Views

Author

Ilya Gutkovskiy, May 02 2018

Keywords

Comments

First differences of A000119.

Links

Crossrefs

Cf. A000045, A000119, A003107, A087897, A238999, A239002.

Programs

  • Mathematica
    nmax = 85; CoefficientList[Series[(1 - x^2) Product[1 + x^Fibonacci[k], {k, 3, 21}], {x, 0, nmax}], x]
Showing 1-3 of 3 results.

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

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