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

1,2

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,...

Links

Table of n, a(n) for n=1..26.

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, ...

Example

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.

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}]

Crossrefs

Cf. A014571, A014572.

Sequence in context: A056815 A127176 A100343 * A041077 A041663 A042825

Adjacent sequences: A085393 A085394 A085395 * A085397 A085398 A085399

Keyword

nonn

Author

Gary W. Adamson, Jun 27 2003

Extensions

Edited by Robert G. Wilson v, Jul 15 2003

Status

approved