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A033090 Indices of incrementally largest terms in the continued fraction for Pi.

%I A033090 #45 Feb 16 2025 08:32:36
%S A033090 1,2,3,5,308,432,28422,156382,267314,453294,11504931,849955263,
%T A033090 2349980289,3588031780,8600404591,15621034283
%N A033090 Indices of incrementally largest terms in the continued fraction for Pi.
%C A033090 This sequence assumes nonstandard indexing of continued fraction terms as [a_1; a_2, a_3, ...]. If you use the actual offset from A001203, corresponding to [a_0; a_1, a_2, ...], you get instead 0, 1, 2, 4, 307, 431, 28421, ... Compare with A033092 versus A224849. - _Jeppe Stig Nielsen_, Dec 14 2019
%H A033090 Syed Fahad, <a href="https://drive.google.com/drive/folders/1--Qh9Xxq1i6oeHnTXzKrQ9FoguHreBKy">30 billion terms of the simple continued fraction of Pi</a>
%H A033090 E. Fontich, C. Simó, and A. Vieiro, <a href="https://doi.org/10.1134/S1560354718060011">On the "hidden" harmonics associated to best approximants due to quasiperiodicity in splitting phenomena</a>, Regular and Chaotic Dynamics (2018), Pleiades Publishing, Vol. 23, Issue 6, 638-653. Also <a href="http://www.maia.ub.es/~carles/moser90.pdf">PDF</a>.
%H A033090 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Pi.html">Pi</a>
%H A033090 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PiContinuedFraction.html">Pi Continued Fraction</a>
%t A033090 With[{s = ContinuedFraction[Pi, 2*10^7]}, Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]] (* _Michael De Vlieger_, Jan 31 2020 *)
%Y A033090 Cf. A033089, A001203.
%K A033090 nonn,hard
%O A033090 1,2
%A A033090 _Eric W. Weisstein_, _Bill Gosper_
%E A033090 a(12) from _Eric W. Weisstein_, Dec 08 2010
%E A033090 a(13) from _Eric W. Weisstein_, Sep 16 2011
%E A033090 a(14) from _Eric W. Weisstein_, Sep 17 2011
%E A033090 a(15) from _Eric W. Weisstein_, Jul 18 2013
%E A033090 a(16) from _Syed Fahad_, Apr 27 2021