A085396 Numerator and denominator sums of convergents to the Thue-Morse constant, 0.412454033...

1, 3, 7, 17, 24, 113, 363, 1928, 4219, 6147, 28807, 63761, 92568, 526601, 23263012, 23789613, 118421464, 142211077, 402843618, 1753585549, 2156429167, 3910014716, 6066443883, 34242234131, 485457721717, 519699955848

Offset: 1

Author

Gary W. Adamson, Jun 27 2003

Keywords

• nonn

Comments

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,...

Examples

			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.
		

Crossrefs

Cf. A014571, A014572.

Programs

Mathematica

mt = 0; Do[ mt = ToString[mt] <> ToString[(10^(2^n) - 1)/9 - ToExpression[mt]], {n, 0, 7}];
d = RealDigits[ N[ ToExpression[mt], 2^7]][[1]];
a = 0; Do[ a = a + N[ d[[n]]/2^(n + 1), 100], {n, 1, 2^7}];
f[n_] := FromContinuedFraction[ ContinuedFraction[a, n]];
Table[ Numerator[ f[n]] + Denominator[ f[n]], {n, 2, 27}]

Formula

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, ...

Extensions

Edited by Robert G. Wilson v, Jul 15 2003