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
Keywords
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
Programs
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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
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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
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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
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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
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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
Comments