A006284 Pierce expansion for Euler's constant.

1, 2, 6, 13, 21, 24, 225, 615, 17450, 23228, 57774, 221361, 522377, 793040, 1706305, 8664354, 19037086, 51965160, 56870701, 124645388, 784244500, 792809072, 3675221276, 42108268014, 53633289500, 56827261536, 67080647365

Offset: 0

Author

N. J. A. Sloane, Jeffrey Shallit

Keywords

• nonn

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Jeffrey Shallit, Some predictable Pierce expansions, Fib. Quart., 22 (1984), 332-335.
• Eric Weisstein's World of Mathematics, Pierce Expansion

Crossrefs

Cf. A006275, A006276, A006283.

Programs

Mathematica

PierceExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@ NestList[{Floor[1/Expand[1 - #[[1]] #[[2]]]], Expand[1 - #[[1]] #[[2]]]} &, {Floor[1/(A - Floor[A])], A - Floor[A]}, n - 1]]; PierceExp[N[EulerGamma, 7!], 25] (* G. C. Greubel, Nov 14 2016 *)

PARI

r=1/Euler;for(n=1,30,r=r/(r-floor(r));print1(floor(r),","))

Formula

If u(0) = exp(1/m), where m is an integer >=1, and u(n+1) = u(n)/frac(u(n)) then floor(u(n)) = m*n. Let u(0)=1/gamma and u(n+1) = u(n)/frac(u(n)) where frac(x) is the fractional part of x, then a(n) = floor(u(n)) - Benoit Cloitre, Mar 09 2004