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.

A294204 Number of partitions of n into distinct Lucas parts (A000032) greater than 1.

Original entry on oeis.org

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

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Author

Ilya Gutkovskiy, Oct 24 2017

Keywords

Comments

Convolution of the sequences A067595 and A033999.

Examples

			a(9) = 2 because we have [7, 2] and [4, 3, 2].

Links

Crossrefs

Cf. A000032, A033999, A067593, A067595, A239002, A294203.

Programs

  • Mathematica
    CoefficientList[Series[(1 + x^2) Product[1 + x^LucasL[k], {k, 2, 15}], {x, 0, 100}], x]

Formula

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

A301653 Expansion of x*(1 + 2*x)/((1 - x)*(1 + x)*(1 - x - x^2)).

Original entry on oeis.org

0, 1, 3, 5, 10, 16, 28, 45, 75, 121, 198, 320, 520, 841, 1363, 2205, 3570, 5776, 9348, 15125, 24475, 39601, 64078, 103680, 167760, 271441, 439203, 710645, 1149850, 1860496, 3010348, 4870845, 7881195, 12752041, 20633238, 33385280, 54018520, 87403801, 141422323, 228826125, 370248450
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 25 2018

Keywords

Comments

Apparently (for n > 0), numbers that have a unique partition into a sum of distinct Lucas numbers (A000204).

Links

Crossrefs

Cf. A000071, A000204, A001622, A003263, A014217, A054770, A294203.

Programs

  • Mathematica
    CoefficientList[Series[x (1 + 2 x)/((1 - x) (1 + x) (1 - x - x^2)) , {x, 0, 40}], x]
LinearRecurrence[{1, 2, -1, -1}, {0, 1, 3, 5}, 41]
Table[LucasL[n + 1] - (3 - (-1)^n)/2, {n, 0, 40}]
Table[Floor[GoldenRatio^(n + 1)] - 1, {n, 0, 40}]
  • PARI
    a(n) = fibonacci(n) + fibonacci(n+2) + ((-1)^n - 3)/2; \\ Altug Alkan, Mar 25 2018

Formula

G.f.: x*(1 + 2*x)/((1 - x)*(1 + x)*(1 - x - x^2)).
a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-4).
a(n) = Lucas(n+1) - (3 - (-1)^n)/2.
a(n) = floor(phi^(n+1)) - 1, where phi = (1 + sqrt(5))/2 is the golden ratio (A001622).
a(n) = Sum_{k>=0} A051601(n-k,k) (conjectured). - Greg Dresden, May 18 2023
Showing 1-2 of 2 results.

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