OFFSET
0,3
COMMENTS
I do not know if this can be modified to give more terms of agreement. - N. J. A. Sloane
From Jon E. Schoenfield, Nov 27 2015: (Start)
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
0.10841015122311136151129081140641509112215809093909
09069019090909059090511902221321990909090909090909
09090909090909090909090909090909090909090909090909
09090909090909090909090909090909090909090909090909
09090909090909090909090909090909090909090909090909
09090909090909090909090909090909090909090909090909
09090909090909090909090909090909090909090909090909
09090909090909090909090909090909090909090909090909
09090909090909090909090909090909090909090909090909
09090909090909090908909048901190716906290416319090
93290909090714119863113213411290313112190412211159
09090909090719034336219011112622211111473522141512
211290113229051159051901390719090909062
and the continued fraction expansion evaluates to
0.10841015122311136151129081140641509112215809093909
090690190909090590905119022213219909090...
(with the digit pair "90" repeating forever), so the two expressions agree through the first 469 digits after the decimal point.
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
0.10841015122311136151129081140641509112215809093909
090690190909090590905119022213219
and the continued fraction expansion evaluates to
0.10841015122311136151129081140641509112215809093909
09070956...
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.
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)
REFERENCES
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.9, p. 443.
LINKS
Pieter Allaart, Stephen Jackson, Taylor Jones and David Lambert, On the existence of Trott numbers, arXiv:2108.03664 [math.NT], 2021.
Pieter Allaart, Stephen Jackson, Taylor Jones, and David Lambert, On the existence of numbers with matching continued fraction and base b expansions, Monatshefte für Mathematik (2023).
Simon Plouffe, The Trott constant.
The Mathematica Journal, Trott Constant.
The Mathematica Journal, Trott's Corner, Finding Trott Constants.
Eric Weisstein's World of Mathematics, Trott Constants.
EXAMPLE
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.
CROSSREFS
KEYWORD
AUTHOR
Simon Plouffe, Michael Trott (mtrott(AT)wolfram.com)
STATUS
approved