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 A219510 #20 Feb 16 2025 08:33:18
%S A219510 1,7,32,97,392,18817,75272,708158977,2832635912,1002978273411373057,
%T A219510 4011913093645492232,2011930833870518011412817828051050497,
%U A219510 8047723335482072045651271312204201992
%N A219510 Pierce expansion of 84 - 48*sqrt(3).
%C A219510 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 = 7. For other cases see A219508 (p = 3), A219509 (p = 5) and A219511 (p = 9).
%H A219510 G. C. Greubel, <a href="/A219510/b219510.txt">Table of n, a(n) for n = 0..20</a>
%H A219510 J. Paradis, P. Viader, and 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 A219510 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 A219510 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PierceExpansion.html">Pierce Expansion</a>
%F A219510 a(2*n) = 2*{(2 + sqrt(3))^(2^n) + (2 - sqrt(3))^(2^n) + 2} for n >= 1.
%F A219510 a(2*n-1) = 1/2*{(2 + sqrt(3))^(2^n) + (2 - sqrt(3))^(2^n)} for n >= 1.
%F A219510 Recurrence equations: a(0) = 1, a(1) = 7 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 A219510 84 - 48*sqrt(3) = 1 - 1/7 + 1/(7*32) - 1/(7*32*97) + 1/(7*32*97*392) - ....
%F A219510 a(2*n) = 8*A002812(n-1)^2 for n >= 1.
%F A219510 a(2*n+1) = A002812(n+1) for n >= 0.
%t A219510 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[84 - 48*Sqrt[3] , 7!], 10] (* _G. C. Greubel_, Nov 15 2016 *)
%o A219510 (PARI) r=(7 + 4*sqrt(3))/12; for(n=1, 10, print(floor(r), ", "); r=r/(r-floor(r))) \\ _G. C. Greubel_, Nov 15 2016
%Y A219510 Cf. A002812, A219508 (p = 3), A219509 (p = 5), A219511 (p = 9).
%K A219510 nonn,easy
%O A219510 0,2
%A A219510 _Peter Bala_, Nov 23 2012