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A006283 Pierce expansion for 1 / Pi.

%I A006283 M3092 #44 Feb 16 2025 08:32:30
%S A006283 3,22,118,383,571,635,70529,375687,399380,575584,699357,1561065,
%T A006283 1795712,194445473,253745996,3199003690,3727084011,6607433185,
%U A006283 16248462801,172940584814,313728984965,796022309187,5348508258636,5962546521072,97497255361780,121347007731845
%N A006283 Pierce expansion for 1 / Pi.
%C A006283 Sequence can be produced with the unit circle: a(1) = number of diameter length arcs in circle rounded down to nearest integer (remainder arc = x_1). a(2) = number of x_1 length arcs in circle rounded down to nearest integer (remainder arc = x_2). a(3) = number of x_2 length arcs in circle rounded down to nearest integer (remainder arc = x_3). And so on ... . - _Peter Woodward_, Sep 08 2016
%D A006283 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006283 G. C. Greubel, <a href="/A006283/b006283.txt">Table of n, a(n) for n = 0..500</a>
%H A006283 Jeffrey Shallit, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/22-4/shallit1.pdf">Some predictable Pierce expansions</a>, Fib. Quart., 22 (1984), 332-335.
%H A006283 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PierceExpansion.html">Pierce Expansion</a>
%e A006283 Let x_0 = 1/Pi = 0.318309886... and a(0) = floor(1/x_0) = 3. Then set x_1 = 1 - a_0*x_0 = 0.0450703..., and a(1) = floor(1/x_1) = 22. Then x_2 = 1 - a_1*x_1 = 0.008452..., and a(2) = floor(1/x2) = 118. - _Michael B. Porter_, Sep 09 2016
%t A006283 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[1/Pi, 8!], 50] (* _G. C. Greubel_, Nov 13 2016 *)
%o A006283 (PARI) default(realprecision, 100000); r=Pi; for(n=1,100,s=(r/(r-floor(r))); print1(floor(r),","); r=s) \\ _Benoit Cloitre_ [amended by _Georg Fischer_, Nov 20 2020]
%Y A006283 Cf. A154956 (analog for 2/Pi).
%K A006283 nonn
%O A006283 0,1
%A A006283 _Jeffrey Shallit_