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 A219509 #22 Feb 16 2025 08:33:18
%S A219509 1,5,24,49,200,4801,19208,46099201,184396808,4250272665676801,
%T A219509 17001090662707208,36129635465198759610694779187201,
%U A219509 144518541860795038442779116748808,2610701117696295981568349760414651575095962187244375364404428801
%N A219509 Pierce expansion of 40 - 16*sqrt(6).
%C A219509 Paradis et al. have determined the Pierce expansion of the quadratic irrationality 2*(p - 1)*(p - sqrt(p^2 - 1)), p a positive integer greater than or equal to 3. This is the case p = 5. For other cases see A219508 (p = 3), A219510 (p = 7) and A219511 (p = 9)
%H A219509 G. C. Greubel, <a href="/A219509/b219509.txt">Table of n, a(n) for n = 0..20</a>
%H A219509 J. Paradis, P. Viader, L. Bibiloni <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/36-2/paradis.pdf">Approximation to quadratic irrationals and their Pierce expansions</a>, The Fibonacci Quarterly, Vol.36 No. 2 (1998) 146-153.
%H A219509 T. A. Pierce, <a href="http://www.jstor.org/stable/2299963">On an algorithm and its use in approximating roots of algebraic equations</a>, Amer. Math. Monthly, Vol. 36 No. 10, (1929) p.523-525.
%H A219509 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PierceExpansion.html">Pierce Expansion</a>
%F A219509 a(2*n+2) = 2*{(5 + 2*sqrt(6))^(2^n) + (5 - 2*sqrt(6))^(2^n) + 2} for n >= 0.
%F A219509 a(2*n+1) = 1/2*{(5 + 2*sqrt(6))^(2^n) + (5 - 2*sqrt(6))^(2^n)} for n >= 0.
%F A219509 Recurrence equations: a(0) = 1, a(1) = 5 and for n >= 1, a(2*n) = 4*(a(2*n-1) + 1) and a(2*n+1) = 2*(a(2*n-1))^2 - 1.
%F A219509 40 - 16*sqrt(6) = sum {n >= 0} 1/product {k = 0..n} a(k) = 1 - 1/5 + 1/(5*24) - 1/(5*24*49) + 1/(5*24*49*200) - ....
%F A219509 a(2*n) = 8*A084765(n-1)^2 for n >= 2.
%F A219509 a(2*n+1) = A084765(n+1) for n >= 0.
%t A219509 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[4*(10 - 4*Sqrt[6]) , 7!], 10] (* _G. C. Greubel_, Nov 14 2016 *)
%o A219509 (PARI) r=(5 + 2*sqrt(6))/8; for(n=1, 10, print(floor(r), ", "); r=r/(r-floor(r))) \\ _G. C. Greubel_, Nov 15 2016
%Y A219509 Cf. A084765, A219508 (p = 3), A219510 (p = 7), A219511 (p = 9).
%K A219509 nonn,easy
%O A219509 0,2
%A A219509 _Peter Bala_, Nov 23 2012