A006284 Pierce expansion for Euler's constant. (Formerly M1593)

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

0,2

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

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

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 *)

Prog

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

Crossrefs

Cf. A006275, A006276, A006283.

Sequence in context: A353588 A353971 A249342 * A048072 A215246 A230537

Adjacent sequences: A006281 A006282 A006283 * A006285 A006286 A006287

Keyword

nonn

Author

N. J. A. Sloane, Jeffrey Shallit

Status

approved