A085394 Numerators of convergents to Thue-Morse constant.

0, 1, 2, 5, 7, 33, 106, 563, 1232, 1795, 8412, 18619, 27031, 153774, 6793087, 6946861, 34580531, 41527392, 117635315, 512068652, 629703967, 1141772619, 1771476586, 9999155549, 141759654272, 151758809821, 7729700145322, 116097260989651

Offset: 1

Author

Gary W. Adamson, Jun 27 2003

Keywords

• frac
• nonn

Examples

			[2,2,2,1,4] = 33/80 = .4125
		

Crossrefs

Cf. A014571, A014572, A085395 (companion denominators).

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]], {n, 1, 28}]

Formula

In continued fraction form, the Thue-Morse constant .4124540336401...; is [2, 2, 2, 1, 4, 3, 5, 2, 1, 4...], with A014572(1) = 2, the first partial quotient. Underneath each term we write the convergents corresponding to the continued fraction: [2] = 1/2, [2, 2] = 2/5, [2, 2, 2] = 5/12 and so on, the convergents being: 1/2, 2/5, 5/12, 7/17, 33/80, 106/257, 563/1365, 1232/2987, 1795/4352, 8412/20395...where the latter = .412454032...

Extensions

Edited by Robert G. Wilson v, Jul 15 2003