A068450 Factorial expansion of sqrt(Pi) = Sum_{n>0} a(n)/n!.

1, 1, 1, 2, 2, 4, 1, 1, 3, 0, 5, 10, 6, 8, 12, 0, 10, 0, 12, 9, 6, 12, 22, 21, 24, 3, 14, 21, 13, 24, 21, 11, 8, 22, 27, 3, 8, 1, 36, 21, 27, 15, 2, 41, 22, 34, 8, 0, 4, 8, 39, 48, 27, 38, 22, 0, 23, 49, 19, 27, 29, 28, 40, 33, 21, 62, 7, 67, 33, 8, 30, 18, 60, 73, 61, 72, 42, 59, 22

Offset: 1

Author

Benoit Cloitre, Mar 10 2002

Keywords

• nonn

Examples

			sqrt(Pi) = 1 + 1/2! + 1/3! + 2/4! + 2/5! + 4/6! + 1/7! + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A075874, A002161 (decimal expansion).

Programs

Magma

SetDefaultRealField(RealField(250));  R:= RealField(); [Floor(Sqrt(Pi(R)))] cat [Floor(Factorial(n)*Sqrt(Pi(R))) - n*Floor(Factorial((n-1))*Sqrt(Pi(R))) : n in [2..30]]; // G. C. Greubel, Mar 21 2018

Mathematica

Table[If[n == 1, Floor[Sqrt[Pi]], Floor[n!*Sqrt[Pi]] - n*Floor[(n - 1)!*Sqrt[Pi]]], {n, 1, 50}] (* G. C. Greubel, Mar 21 2018 *)

PARI

default(realprecision, 250); for(n=1,30, print1(if(n==1, floor(sqrt(Pi)), floor(n!*sqrt(Pi)) - n*floor((n-1)!*sqrt(Pi))), ", ")) \\ G. C. Greubel, Mar 21 2018

PARI

vector(30,n,if(n>1,t=t%1*n,t=sqrt(Pi))\1) \\ M. F. Hasler, Nov 25 2018

Sage

def A068450(n):
    if (n==1): return floor(sqrt(pi))
    else: return expand(floor(factorial(n)*sqrt(pi)) - n*floor(factorial(n-1)*sqrt(pi)))
[A068450(n) for n in (1..80)] # G. C. Greubel, Nov 27 2018

Extensions

Keyword cons removed by R. J. Mathar, Jul 23 2009