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Numerator and denominator sums of convergents to the Thue-Morse constant, 0.412454033...
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%I #11 Feb 06 2022 10:07:34

%S 1,3,7,17,24,113,363,1928,4219,6147,28807,63761,92568,526601,23263012,

%T 23789613,118421464,142211077,402843618,1753585549,2156429167,

%U 3910014716,6066443883,34242234131,485457721717,519699955848

%N Numerator and denominator sums of convergents to the Thue-Morse constant, 0.412454033...

%C Let k = 0.412454..., then A085396(n)/A085394(n) [i.e., (numerator + denominator)/(numerator)] converges upon 3.424512... as n approaches infinity, where 3.424... = (k+1)/k. A085396(n)/A085395(n) [i.e., (numerator + denominator)/(denominator)], converges upon k+1, = 1.412454... Check: A085396(6)/A085394(6) = 363/106 = 3.4245...; while A085396(6)/A085395(6) = 393/257 = 1.41245... The constants (k+1) and (k+1)/k are generators for the Beatty pairs for the Thue-Morse constant, where the pairs are [(n*(k+1), (n*(k+1)/k], n = 1,2,3,...

%F a(n) = A085394(n) + A085395(n) = numerator and denominator sums for convergents of 0.412454..., the convergents being 1/2, 2/5, 5/12, 7/17, 33/80, 106/257, 563/1365, 1232/2987, 1795/4352, 8412/20395, ...

%e Convergents to the Thue-Morse constant 0.4124540336... are derived from continued fraction form shown in A014572, starting with A014572(1) = 2; then 0.412454... = [2, 2, 2, 1, 4, 3, 5, 2, 1, ...] (A014572). Example [2] = 1/2, [2,2] = 2/5, [2,2,2] = 5/12 and so on.

%t mt = 0; Do[ mt = ToString[mt] <> ToString[(10^(2^n) - 1)/9 - ToExpression[mt]], {n, 0, 7}];

%t d = RealDigits[ N[ ToExpression[mt], 2^7]][[1]];

%t a = 0; Do[ a = a + N[ d[[n]]/2^(n + 1), 100], {n, 1, 2^7}];

%t f[n_] := FromContinuedFraction[ ContinuedFraction[a, n]];

%t Table[ Numerator[ f[n]] + Denominator[ f[n]], {n, 2, 27}]

%Y Cf. A014571, A014572.

%K nonn

%O 1,2

%A _Gary W. Adamson_, Jun 27 2003

%E Edited by _Robert G. Wilson v_, Jul 15 2003