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

1,1

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

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

Formula

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

Example

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

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 *)

Prog

(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
(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
(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

Crossrefs

Essentially same as A007514.

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

Cf. A068452-A068464.

Sequence in context: A063691 A359967 A284281 * A181634 A230652 A230661

Adjacent sequences: A075871 A075872 A075873 * A075875 A075876 A075877

Keyword

nonn

Author

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

Status

approved