A068452 Pi^2 = Sum_{n>=1} a(n)/n!.

9, 1, 2, 0, 4, 2, 0, 6, 4, 0, 4, 11, 6, 4, 14, 8, 12, 6, 18, 12, 12, 14, 13, 2, 7, 20, 12, 2, 16, 21, 25, 26, 29, 19, 7, 3, 20, 3, 38, 7, 12, 19, 37, 1, 23, 32, 19, 32, 38, 45, 45, 27, 44, 34, 14, 49, 35, 29, 30, 57, 57, 18, 56, 48, 33, 19, 44, 35, 12, 56, 28, 38, 64, 35, 10, 45, 35, 0

Offset: 1

Author

Benoit Cloitre, Mar 10 2002

Keywords

• nonn

Comments

For the fractional part, this corresponds to the factoradic (or factorial base, or harmonic) expansion, but the integer part 9 = 3! + 2! + 1! would be [1, 1, 1] in factorial base, cf. A007623(9) = 111. - M. F. Hasler, Nov 27 2018

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Vincenzo Librandi)
• Eric Weisstein's World of Mathematics, Harmonic Expansion.
• Wikipedia, Factorial number system: Fractional values
• Index to sequences related to factorial base representation of noninteger constants.

Crossrefs

Cf. A002388 (decimal expansion of Pi^2).

Similar expansions: A068450 (sqrt(Pi)), A075874 (Pi), A007514 (different variant for Pi).

Programs

Magma

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

Maple

fexp := proc(x) local xres,a,n ; xres := x ; a := [] ; for n from 1 to 100 do a := [op(a),floor(n!*xres)] ; xres := xres-op(-1,a)/n! ; od: a ; end: Digits := 400 ; fexp(evalf(Pi^2)) ; Digits := 600 ; fexp(evalf(Pi^2)) ; # R. J. Mathar, Sep 30 2008

Mathematica

p=N[Pi, 10000]^2; Do[k=Floor[p n!]; p=p - k / n!; Print[k], {n, 1000}] (* Vincenzo Librandi, Nov 24 2018 *)
With[{b = Pi^2}, Table[If[n == 1, Floor[b], Floor[n!*b] -n*Floor[(n- 1)!*b]], {n, 1, 100}]] (* G. C. Greubel, Nov 26 2018 *)

PARI

default(realprecision, 250); b = Pi^2; for(n=1, 80, print1(if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b)), ", ")) \\ G. C. Greubel, Nov 26 2018

PARI

A068452(N=90, c=precision(Pi^2,logint(N!,10)))=vector(N, n, if(n>1, c=c%1*n, c)\1) \\ M. F. Hasler, Nov 27 2018

Sage

def A068452(n):
    if (n==1): return floor(pi^2)
    else: return expand(floor(factorial(n)*pi^2) - n*floor(factorial(n-1)*pi^2))
[A068452(n) for n in (1..80)] # G. C. Greubel, Nov 26 2018

Extensions

Corrected beginning at 3rd term by R. J. Mathar, Sep 30 2008