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%I #55 May 31 2023 09:07:45
%S 1,0,8,4,1,0,1,5,1,2,2,3,1,1,1,3,6,1,5,1,1,2,9,0,8,1,1,4,0,6,4,1,5,0,
%T 9,1,1,2,2,1,5,8,0,9,0,9,3,9,0,9,0,9,1
%N Decimal expansion of Michael Trott's constant: continued fraction expansion (allowing 0's) begins in same way as decimal expansion.
%C I do not know if this can be modified to give more terms of agreement. - _N. J. A. Sloane_
%C From _Jon E. Schoenfield_, Nov 27 2015: (Start)
%C Removing the final digit listed in the Data section (i.e., a(-52)=1) allows the sequence to be extended so as to yield an arbitrarily large number of terms of agreement by applying the procedure described at the "Extending" link at A114376. The best result that can be obtained using a practical number of terms occurs at 639 terms, at which point the decimal expansion is
%C 0.10841015122311136151129081140641509112215809093909
%C 09069019090909059090511902221321990909090909090909
%C 09090909090909090909090909090909090909090909090909
%C 09090909090909090909090909090909090909090909090909
%C 09090909090909090909090909090909090909090909090909
%C 09090909090909090909090909090909090909090909090909
%C 09090909090909090909090909090909090909090909090909
%C 09090909090909090909090909090909090909090909090909
%C 09090909090909090909090909090909090909090909090909
%C 09090909090909090908909048901190716906290416319090
%C 93290909090714119863113213411290313112190412211159
%C 09090909090719034336219011112622211111473522141512
%C 211290113229051159051901390719090909062
%C and the continued fraction expansion evaluates to
%C 0.10841015122311136151129081140641509112215809093909
%C 090690190909090590905119022213219909090...
%C (with the digit pair "90" repeating forever), so the two expressions agree through the first 469 digits after the decimal point.
%C The long string of digit pairs "90" beginning immediately after the 83rd digit after the decimal point arises from the very close agreement of the decimal expansion and the continued fraction expansion at that point; at 83 terms, the decimal expansion is exactly
%C 0.10841015122311136151129081140641509112215809093909
%C 090690190909090590905119022213219
%C and the continued fraction expansion evaluates to
%C 0.10841015122311136151129081140641509112215809093909
%C 09070956...
%C yielding agreement through the first 53 digits after the decimal point. After this, the ratio (digits of agreement) / (number of terms) drops rapidly, not returning to similar levels again until the number of terms reaches 639.
%C While it is theoretically possible to extend this sequence arbitrarily far using the same procedure, it is impractical to do so; the agreement using 639 terms is so extremely close that the number of consecutive digit pairs "90" (i.e., term pairs (9, 0)) that would immediately follow the 639th term would exceed 5*10^301. (End)
%H Pieter Allaart, Stephen Jackson, Taylor Jones and David Lambert, <a href="https://arxiv.org/abs/2108.03664v1">On the existence of Trott numbers</a>, arXiv:2108.03664 [math.NT], 2021.
%H Pieter Allaart, Stephen Jackson, Taylor Jones, and David Lambert, <a href="https://doi.org/10.1007/s00605-023-01873-8">On the existence of numbers with matching continued fraction and base b expansions</a>, Monatshefte für Mathematik (2023).
%H Simon Plouffe, <a href="http://www.plouffe.fr/simon/constants/trott.txt">The Trott constant</a>
%H The Mathematica Journal, <a href="http://www.mathematica-journal.com/2006/09/finding-trott-constants/">Trott Constant</a>
%H The Mathematica Journal, Trott's Corner, <a href="http://www.mathematica-journal.com/data/uploads/2012/05/Corner10-2.pdf">Finding Trott Constants</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TrottConstants.html">Trott Constants</a>
%e Let {a,b,c,d,...} mean a+1/(b+1/(c+1/(d+...))). Then {0,1,0,...,9,0,9,1} = 0 + 1/(1 + 1/(0 + 1/ (8 + 1/ (4 + 1/ (1 + 1/ (0 + 1/ (1 + 1/ (5 + 1/(1 + 1/ (2 + 1/ (2 + 1/ (3 + 1/ (1 + 1/(1 + 1/(1 + 1/(3 + 1/(6 + 1/(1 + 1/(5 + 1/(1 + 1/(1 + 1/(2 + 1/(9 + 1/(0 +1/(8 + 1/(1 + 1/(1 +1/(4 + 1/(0 + 1/(6 + 1/(4 + 1/(1 + 1/(5 + 1/(0 + 1/(9 +1/(1 + 1/(1 + 1/(2 + 1/(2 + 1/(1 + 1/(5 + 1/(8 + 1/(0 + 1/(9 + 1/(0 +1/(9 + 1/(3 + 1/(9 + 1/(0 + 1/(9 + 1/1)))))))))))))))))))))))))))))))))))))))))))))))))) = 17928273845270692 / 165374493467625219 = 0.108410151223111361511290811406414793... and the digits agree to 32 places.
%Y Cf. A039663, A091694, A113307, A114376, A169670.
%K nonn,base,more,cons
%O 0,3
%A _Simon Plouffe_, Michael Trott (mtrott(AT)wolfram.com)
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