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.

A131443 Numerators of n-th approximation of factorial (also called harmonic) expansion of the golden section phi:=(1+sqrt(5))/2.

Original entry on oeis.org

1, 3, 3, 19, 97, 97, 453, 65239, 36697, 5871521, 3075559, 775040869, 10075531303, 10075531303, 2115861573641, 2115861573641, 22135167231937, 941750751322411, 39365181405276781, 3936518140527678109
Offset: 1

Views

Author

Wolfdieter Lang, Aug 07 2007

Keywords

Comments

Denominators are given in A131444. Rationals in lowest terms.

Examples

			Rationals r(n):[1, 3/2, 3/2, 19/12, 97/60, 97/60, 453/280, 65239/40320, ...].

Links

Formula

a(n)=numerator(r(n)), with r(n):=sum(b(k)/k!,n=1..n) with b(k):=A068451(k ) (factorial expansion of phi).

A131444 Denominators of n-th approximation of factorial (also called harmonic) expansion of the golden ratio phi (A001622).

Original entry on oeis.org

1, 2, 2, 12, 60, 60, 280, 40320, 22680, 3628800, 1900800, 479001600, 6227020800, 6227020800, 1307674368000, 1307674368000, 13680285696000, 582033973248000, 24329020081766400, 2432902008176640000
Offset: 1

Views

Author

Wolfdieter Lang, Aug 07 2007

Keywords

Comments

For the rationals r(n) see the W. Lang link found under A131443.

Examples

			Rationals r(n):[1, 3/2, 3/2, 19/12, 97/60, 97/60, 453/280, 65239/40320,...].

Crossrefs

Cf. A001622, A068451, A131443 (numerators).

Formula

a(n) = denominator(r(n)), with r(n) = Sum_{k=1..n} b(k)/k!, with b(k) = A068451(k).

A322119 Factorial expansion of (1-sqrt(5))/2 = Sum_{n>=1} a(n)/n!.

Original entry on oeis.org

-1, 0, 2, 1, 0, 5, 0, 0, 7, 8, 2, 10, 6, 13, 3, 15, 6, 12, 12, 10, 5, 1, 12, 8, 23, 7, 21, 14, 19, 29, 17, 16, 30, 6, 6, 33, 4, 1, 27, 35, 6, 4, 42, 39, 12, 35, 42, 43, 16, 3, 11, 14, 50, 33, 27, 47, 2, 30, 13, 50, 34, 43, 3, 63, 42, 2, 25, 13, 3, 8, 25, 20, 11, 42, 6, 27, 42, 38, 7, 20
Offset: 1

Views

Author

G. C. Greubel, Nov 26 2018

Keywords

Comments

This expansion can also be considered as the expansion of -1/(golden ratio).

Examples

			(1-sqrt(5))/2 = -1 + 2/3! + 1/4! + 5/6! + 7/9! + 8/10! + 2/11! + ...

Links

Crossrefs

Cf. A001622, A068451, A094214 (decimal expansion, negated).

Programs

  • Magma
    SetDefaultRealField(RealField(250));  [Floor((1-Sqrt(5))/2)] cat [Floor(Factorial(n)*(1-Sqrt(5))/2) - n*Floor(Factorial((n-1))*(1-Sqrt(5))/2) : n in [2..80]];
  • Mathematica
    With[{b = -1/GoldenRatio}, Table[If[n == 1, Floor[b], Floor[n!*b] - n*Floor[(n - 1)!*b]], {n, 1, 100}]]
  • PARI
    default(realprecision, 250); b = (1-sqrt(5))/2; for(n=1, 80, print1(if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b)), ", "))
  • Sage
    def a(n):
    if (n==1): return floor(-1/golden_ratio)
    else: return expand(floor(factorial(n)*(-1/golden_ratio)) - n*floor(factorial(n-1)*(-1/golden_ratio)))
[a(n) for n in (1..80)]
Showing 1-3 of 3 results.

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

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