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 A317557 Number of binary digits to which the n-th convergent of the continued fraction expansion of log(2) matches the correct value. 3
 0, -1, 3, 6, 9, 13, 14, 17, 19, 20, 23, 20, 25, 20, 33, 37, 35, 38, 41, 43, 45, 43, 47, 48, 52, 54, 58, 61, 68, 70, 74, 77, 78, 81, 86, 89, 92, 93, 92, 99, 105, 109, 113, 116, 118, 121, 127, 133, 136, 135, 139, 141, 145, 149, 154, 159, 161, 165, 171, 173, 172, 180 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Binary expansion of log(2) in A068426. For number of correct decimal digits see A317558. For the similar case of number of correct binary digits of Pi see A305879. The denominator of the k-th convergent obtained from a continued fraction satisfying the Gauss-Kuzmin distribution will tend to exp(k*A100199), A100199 being the inverse of Lévy's constant; the error between the k-th convergent and the constant itself tends to exp(-2*k*A100199), or in binary digits 2*k*A100199/log(2) bits after the binary point. The sequence for quaternary digits is obtained by floor(a(n)/2), the sequence for octal digits is obtained by floor(a(n)/3), and the sequence for hexadecimal digits is obtained by floor(a(n)/4). LINKS A.H.M. Smeets, Table of n, a(n) for n = 1..20000 FORMULA Lim_{n -> oo} a(n)/n = 2*log(A086702)/log(2) = 2*A100199/log(2) = 2*A305607. EXAMPLE n convergent binary expansion a(n) == ============ ============================= ==== 1 0 / 1 0.0 0 2 1 / 1 1.0 -1 3 2 / 3 0.1010... 3 4 7 / 10 0.1011001... 6 5 9 / 13 0.1011000100... 9 6 61 / 88 0.10110001011101... 13 7 192 / 277 0.101100010111000... 14 8 253 / 365 0.101100010111001001... 17 9 445 / 642 0.10110001011100100000... 19 10 1143 / 1649 0.101100010111001000011... 20 oo lim = log(2) 0.101100010111001000010111... -- MATHEMATICA a[n_] := Block[{k = 1, a = RealDigits[ Log@2, 2, 4 + 10][], b = RealDigits[ FromContinuedFraction@ ContinuedFraction[Log@2, n + 1], 2, 4n + 10][]}, While[ a[[k]] == b[[k]], k++]; k - 1]; a = 0; a = -1; Array[a, 61] (* Robert G. Wilson v, Aug 09 2018 *) CROSSREFS Cf. A016730, A068426, A086702, A100199, A305607, A317558. Sequence in context: A065811 A061514 A078559 * A338763 A159908 A236761 Adjacent sequences: A317554 A317555 A317556 * A317558 A317559 A317560 KEYWORD sign,base AUTHOR A.H.M. Smeets, Jul 31 2018 EXTENSIONS a(40) onward from Robert G. Wilson v, Aug 09 2018 STATUS approved

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Last modified October 1 02:41 EDT 2023. Contains 365812 sequences. (Running on oeis4.)