A089078 Continued fraction for sqrt(2) + sqrt(3).

3, 6, 1, 5, 7, 1, 1, 4, 1, 38, 43, 1, 3, 2, 1, 1, 1, 1, 2, 4, 1, 4, 5, 1, 5, 1, 7, 22, 2, 5, 1, 1, 2, 1, 1, 31, 2, 1, 1, 3, 1, 44, 1, 89, 1, 8, 5, 2, 5, 1, 1, 4, 2, 8, 1, 17, 1, 4, 3, 4, 3, 2, 1, 1, 4, 2, 1, 9, 1, 15, 13, 1, 39, 20, 2, 152, 3, 2, 4, 1, 30, 1, 3, 1, 2, 1, 2, 16, 3, 24, 1, 9, 1, 172, 3, 1, 1

Offset

0,1

Comments

This is the most natural example of the fact that the sum of two periodic continued fractions need not have a periodic continued fraction.

a(n) is the numbers of squares removed at stage n of the continued-fraction partitioning of a rectangle of length L and width W satisfying W=L*sqrt(8); see A188640. - Clark Kimberling, Apr 13 2011

Links

G. C. Greubel, Table of n, a(n) for n = 0..5000

G. Xiao, Contfrac

Mathematica

r = 8^(1/2); t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
ContinuedFraction[t, 120]
ContinuedFraction[Sqrt[2]+Sqrt[3], 100] (* Harvey P. Dale, Aug 17 2019 *)

Prog

(PARI) contfrac(sqrt(2)+sqrt(3)) \\ Michel Marcus, Mar 12 2017

Crossrefs

Cf. A135611.

Sequence in context: A175032 A078768 A307281 * A355072 A134804 A145389

Adjacent sequences: A089075 A089076 A089077 * A089079 A089080 A089081

Keyword

cofr,nonn

Author

Jeppe Stig Nielsen, Dec 04 2003

Status

approved