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 A188725 #19 Oct 02 2022 22:57:22 %S A188725 6,4,3,8,5,0,0,9,6,3,0,6,5,4,0,8,3,9,7,2,2,3,2,3,2,5,6,3,5,9,4,6,9,1, %T A188725 7,2,9,2,6,2,1,6,6,5,4,0,8,1,3,2,6,1,5,2,5,6,1,0,6,5,1,7,3,2,5,8,9,5, %U A188725 9,2,1,2,6,3,3,4,3,7,5,1,1,6,9,3,8,6,9,6,6,9,2,7,7,2,1,5,3,0,9,8,5,0,0,3,9,3,0,2,8,1,2,1,5,8,5,8,7,0,2,3,1,6,7,6,5,3,0,9,1,5 %N A188725 Decimal expansion of shape of a (2*Pi)-extension rectangle; shape = Pi + sqrt(1 + Pi^2). %C A188725 See A188640 for definitions of shape and r-extension rectangle. Briefly, an r-extension rectangle is composed of two rectangles of shape r. %C A188725 A 2*Pi-extension rectangle matches the continued fraction [6,2,3,1,1,3,2,1,16,47,...] of the shape L/W = Pi + sqrt(1 + Pi^2). This is analogous to the matching of a golden rectangle to the continued fraction [1,1,1,1,1,1,1,...]. Specifically, for the (2*Pi)-extension rectangle, 6 squares are removed first, then 2 squares, then 3 squares, then 1 square, then 1 square, ..., so that the original rectangle is partitioned into an infinite collection of squares. %H A188725 G. C. Greubel, <a href="/A188725/b188725.txt">Table of n, a(n) for n = 1..10000</a> %H A188725 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a> %e A188725 6.4385009630654083972232325635946917292621665408132615256106... %p A188725 evalf(Pi+sqrt(1+Pi^2),140); # _Muniru A Asiru_, Nov 01 2018 %t A188725 r = 2*Pi; t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t] %t A188725 N[t, 130] %t A188725 RealDigits[N[t, 130]][[1]] (* A188725 *) %t A188725 ContinuedFraction[t, 120] (* A188726 *) %t A188725 RealDigits[Pi + Sqrt[1 + Pi^2], 10, 100][[1]] (* _G. C. Greubel_, Oct 31 2018 *) %o A188725 (PARI) default(realprecision, 100); Pi + sqrt(1 + Pi^2) \\ _G. C. Greubel_, Oct 31 2018 %o A188725 (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); Pi(R) + Sqrt(1 + Pi(R)^2); // _G. C. Greubel_, Oct 31 2018 %Y A188725 Cf. A188640, A188726. %K A188725 nonn,cons %O A188725 1,1 %A A188725 _Clark Kimberling_, Apr 09 2011