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.

A068464 Factorial expansion of Gamma(1/4) = Sum_{n>=1} a(n)/n! with largest possible a(n), and Gamma = Euler's gamma function.

Original entry on oeis.org

3, 1, 0, 3, 0, 0, 3, 0, 5, 3, 2, 7, 0, 2, 8, 9, 16, 3, 1, 15, 18, 8, 20, 7, 23, 8, 10, 11, 28, 29, 24, 30, 3, 16, 10, 8, 31, 11, 30, 35, 5, 5, 38, 32, 31, 42, 13, 17, 43, 3, 41, 27, 1, 14, 26, 52, 38, 22, 55, 46, 6, 35, 46, 34, 24, 52, 52, 64, 62, 25, 46, 56, 3, 71, 70, 22, 53, 63, 53
Offset: 1

Views

Author

Benoit Cloitre, Mar 10 2002

Keywords

Examples

			Gamma(1/4) = A068466 = 3.6256099... = 3/1! + 1/2! + 0 + 3/4! + 0 + 0 + 3/7! + 0 + 5/9! + 3/10! + 2/11! + ... - _M. F. Hasler_, Nov 26 2018

Links

Crossrefs

Cf. A007514, A068466 (decimal expansion), A068463.

Programs

  • Magma
    SetDefaultRealField(RealField(250));  [Floor(Gamma(1/4))] cat [Floor(Factorial(n)*Gamma(1/4)) - n*Floor(Factorial((n-1))*Gamma(1/4)) : n in [2..80]]; // G. C. Greubel, Nov 27 2018
  • Mathematica
    r:= Gamma[1/4]; Table[If[n == 1, Floor[r], Floor[n!*r]- n*Floor[(n-1)!*r] ], {n,1,100}] (* G. C. Greubel, Mar 29 2018 *)
  • PARI
    default(realprecision, 250); b = gamma(1/4); for(n=1, 80, print1(if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b)), ", ")) \\ G. C. Greubel, Mar 29 2018
  • PARI
    A068464(N=90,c=gamma(precision(.25,logint(N!,10)+1)))=vector(N,n,if(n>1,c=c%1*n,c)\1) \\ - M. F. Hasler, Nov 26 2018
  • Sage
    def A068464(n):
    if (n==1): return floor(gamma(1/4))
    else: return expand(floor(factorial(n)*gamma(1/4)) - n*floor(factorial(n-1)*gamma(1/4)))
[A068464(n) for n in (1..80)] # G. C. Greubel, Nov 27 2018

Formula

a(n) = floor(n!*Gamma(1/4)) - n*floor((n-1)!*Gamma(1/4)), for n > 1. - M. F. Hasler, Nov 26 2018

A322508 Factorial expansion of Gamma(1/3) = Sum_{n>=1} a(n)/n!.

Original entry on oeis.org

2, 1, 1, 0, 1, 2, 5, 6, 7, 2, 1, 8, 5, 7, 9, 12, 13, 10, 10, 13, 17, 18, 5, 1, 6, 3, 26, 13, 20, 29, 8, 31, 27, 19, 21, 27, 5, 14, 12, 3, 9, 37, 34, 40, 14, 29, 35, 12, 27, 4, 36, 22, 24, 11, 31, 37, 12, 5, 47, 14, 22, 18, 51, 20, 51, 4, 15, 54, 61, 26, 55, 2, 6, 73, 7, 17, 66, 54, 27
Offset: 1

Views

Author

G. C. Greubel, Dec 12 2018

Keywords

Examples

			Gamma(1/3) = 2 + 1/2! + 1/3! + 0/4! + 1/5! + 2/6! + 5/7! + 6/8! + ...

Links

Crossrefs

Cf. A073005 (decimal expansion), A030651 (continued fraction).
Cf. A068463 (Gamma(3/4)), A068464 (Gamma(1/4)), A322509 (Gamma(2/3)).

Programs

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

A322509 Factorial expansion of Gamma(2/3) = Sum_{n>=1} a(n)/n!.

Original entry on oeis.org

1, 0, 2, 0, 2, 2, 6, 6, 0, 3, 1, 11, 7, 6, 6, 14, 1, 8, 12, 15, 8, 17, 8, 1, 13, 15, 3, 4, 10, 16, 25, 1, 25, 22, 6, 3, 19, 17, 8, 10, 25, 37, 29, 17, 35, 19, 24, 25, 30, 31, 4, 7, 51, 49, 14, 51, 45, 54, 0, 26, 34, 41, 56, 57, 16, 15, 63, 4, 51, 42, 13, 35, 12, 15, 66, 22, 13, 43, 14, 78
Offset: 1

Views

Author

G. C. Greubel, Dec 12 2018

Keywords

Examples

			Gamma(2/3) = 1 + 0/2! + 2/3! + 0/4! + 2/5! + 2/6! + 6/7! + 6/8! + ...

Links

Crossrefs

Cf. A073006 (decimal expansion), A030652 (continued fraction).
Cf. A068463 (Gamma(3/4)), A068464 (Gamma(1/4)), A322508 (Gamma(1/3)).

Programs

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

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