This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A091831 #26 Feb 16 2025 08:32:52
%S A091831 1,3,8,33,35,39201,39203,60245508192801,60245508192803,
%T A091831 218662352649181293830957829984632156775201,
%U A091831 218662352649181293830957829984632156775203
%N A091831 Pierce expansion of 1/sqrt(2).
%C A091831 If u(0)=exp(1/m) m integer>1 and u(n+1)=u(n)/frac(u(n)) then floor(u(n))=m*n.
%H A091831 G. C. Greubel, <a href="/A091831/b091831.txt">Table of n, a(n) for n = 0..14</a>
%H A091831 P. Erdős and Jeffrey Shallit, <a href="http://www.numdam.org/item?id=JTNB_1991__3_1_43_0">New bounds on the length of finite Pierce and Engel series</a>, Sem. Théor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.
%H A091831 Vlado Keselj, <a href="https://cs.uwaterloo.ca/research/tr/1996/21/cs-96-21.pdf">Length of Finite Pierce Series: Theoretical Analysis and Numerical Computations </a>.
%H A091831 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 A091831 Pelegrí Viader, Lluís Bibiloni, Jaume Paradís, <a href="http://dx.doi.org/10.2139/ssrn.145561">On a problem of Alfred Renyi</a>, Economics Working Paper No. 340.
%H A091831 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PierceExpansion.html">Pierce Expansion</a>
%F A091831 Let u(0)=sqrt(2) and u(n+1)=u(n)/frac(u(n)) where frac(x) is the fractional part of x, then a(n)=floor(u(n)).
%F A091831 1/sqrt(2)= 1/a(1) - 1/a(1)/a(2) + 1/a(1)/a(2)/a(3) - 1/a(1)/a(2)/a(3)/a(4)...
%F A091831 limit n -> infinity a(n)^(1/n) = e.
%t A091831 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[2^(-1/2), 7!], 17] (* _G. C. Greubel_, Nov 13 2016 *)
%o A091831 (PARI) r=sqrt(2);for(n=1,10,r=r/(r-floor(r));print1(floor(r),","))
%Y A091831 Cf. A006275, A006276, A006283.
%Y A091831 Cf. A006784 (Pierce expansion definition), A028254
%K A091831 nonn
%O A091831 0,2
%A A091831 _Benoit Cloitre_, Mar 09 2004