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 A039662 Decimal expansion of Michael Trott's constant: continued fraction expansion (allowing 0's) begins in same way as decimal expansion. 6
 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, 9, 1, 1, 2, 2, 1, 5, 8, 0, 9, 0, 9, 3, 9, 0, 9, 0, 9, 1 (list; constant; graph; refs; listen; history; text; internal format)
 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) LINKS 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 Cf. A039663, A091694, A113307, A114376, A169670. Sequence in context: A199285 A049255 A028577 * A228496 A259616 A247036 Adjacent sequences:  A039659 A039660 A039661 * A039663 A039664 A039665 KEYWORD nonn,more,cons AUTHOR Simon Plouffe, Michael Trott (mtrott(AT)wolfram.com) STATUS approved

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