A075874 Pi = Sum_{n >= 1} a(n)/n!, with largest possible a(n).

3, 0, 0, 3, 1, 5, 6, 5, 0, 1, 4, 7, 8, 0, 6, 7, 10, 7, 10, 4, 10, 6, 16, 1, 11, 20, 3, 18, 12, 9, 13, 18, 21, 14, 34, 27, 11, 27, 33, 36, 18, 5, 18, 5, 23, 39, 1, 10, 42, 28, 17, 20, 51, 8, 42, 47, 0, 27, 23, 16, 52, 32, 52, 53, 24, 43, 61, 64, 18, 17, 11, 0, 53, 14, 62

Offset: 1

Author

N. J. A. Sloane, Robert G. Wilson v, Nov 02 2001 and Oct 20 2002

Keywords

• nonn

Comments

What is meant is the expansion in the factorial number system, cf. links. The formula itself is not sufficient to define the terms uniquely: a(n) can be decreased by any amount x if x*(n+1) is added to a(n+1). - M. F. Hasler, Nov 26 2018

Examples

			Pi = 3/1! + 0/2! + 0/3! + 3/4! + 1/5! + ...
		

Links

• Hans Havermann, Table of n, a(n) for n = 1..10000
• D. E. Knuth, The Art of Computer Programming, Vol.2, 3rd ed., Addison-Wesley, 2014, ISBN 978-0321635761, p.209.
• Eric Weisstein's World of Mathematics, Harmonic Expansion
• Wikipedia, Factorial number system

Crossrefs

Essentially same as A007514.

Pi in base n: A004601 to A004608, A000796, A068436 to A068440, A062964.

Cf. A068452-A068464.

Programs

Magma

SetDefaultRealField(RealField(250)); R:=RealField(); [Floor(Pi(R))] cat [Floor(Factorial(n)*Pi(R)) - n*Floor(Factorial((n-1))*Pi(R)) : n in [2..80]]; // G. C. Greubel, Nov 26 2018

Maple

Digits := 120; M := proc(a,n) local i,b,c; b := a; c := [ floor(b) ]; for i from 1 to n-1 do b := b-c[ i ]/i!; c := [ op(c), floor(b*(i+1)!) ]; od; c; end: t1 := M(Pi,100); A075874 := n->t1[n+1];

Mathematica

p = N[Pi, 1000]; Do[k = Floor[p*n! ]; p = p - k/n!; Print[k], {n, 1, 75}]
With[{b = Pi}, Table[If[n == 1, Floor[b], Floor[n!*b] - n*Floor[(n - 1)!*b]], {n, 1, 100}]] (* G. C. Greubel, Nov 26 2018 *)

PARI

x=Pi;vector(floor((y->y/log(y))(default(realprecision))),n,t=n!;k=floor(x*t);x-=k/t;k) \\ Charles R Greathouse IV, Jul 15 2011

PARI

vector(30,n,if(n>1,t=t%1*n,t=Pi)\1) \\ Increase realprecision (e.g., \p500) to compute more terms. - M. F. Hasler, Nov 25 2018

PARI

default(realprecision, 250); b = Pi; for(n=1, 80, print1(if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b)), ", ")) \\ G. C. Greubel, Nov 26 2018

Sage

def A075874(n):
    if (n==1): return floor(pi)
    else: return expand(floor(factorial(n)*pi) - n*floor(factorial(n-1)*pi))
[A075874(n) for n in (1..80)] # G. C. Greubel, Nov 26 2018

Formula

a(1)=3; for n >= 2, a(n) = floor(n!*Pi) - n*floor((n-1)!*Pi). - Benoit Cloitre, Mar 10 2002