A188485 Decimal expansion of (3+sqrt(17))/4, which has periodic continued fractions [1,1,3,1,1,3,1,1,3,...] and [3/2, 3, 3/2, 3, 3/2, ...].

1, 7, 8, 0, 7, 7, 6, 4, 0, 6, 4, 0, 4, 4, 1, 5, 1, 3, 7, 4, 5, 5, 3, 5, 2, 4, 6, 3, 9, 9, 3, 5, 1, 9, 2, 5, 6, 2, 8, 6, 7, 9, 9, 8, 0, 6, 3, 4, 3, 4, 0, 5, 1, 0, 8, 5, 9, 9, 6, 5, 8, 3, 9, 3, 2, 7, 3, 7, 3, 8, 5, 8, 6, 5, 8, 4, 4, 0, 5, 3, 9, 8, 3, 9, 6, 9, 6, 5, 9, 1, 2, 7, 0, 2, 6, 7, 1, 0, 7, 4, 1, 7, 1, 1, 3, 6, 0, 1, 0, 2, 3, 4, 8, 0, 3, 5, 3, 5, 4, 0

Offset: 1

Author

Clark Kimberling, May 05 2011

Keywords

• nonn
• cons

Comments

Let R denote a rectangle whose shape (i.e., length/width) is (3+sqrt(17))/3. This rectangle can be partitioned into squares in a manner that matches the continued fraction [1,1,3,1,1,3,1,1,3,...]. It can also be partitioned into rectangles of shape 3/2 and 3 so as to match the continued fraction [3/2, 3, 3/2, 3, 3/2, ...]. For details, see A188635.

Apart from the second digit the same as A188934. - R. J. Mathar, May 16 2011

Equivalent to the infinite continued fraction with denominators [1; 2, 1, 2, 1, ...] and numerators [2, 1, 2, ...], also expressible as 1+2/(2+1/(1+2/(2+1/...))). - Matthew A. Niemiro, Dec 13 2019

Examples

			1.780776406404415137455352463993519256287...
		

Links

• J. S. Brauchart, P. D. Dragnev, E. B. Saff, An Electrostatics Problem on the Sphere Arising from a Nearby Point Charge, arXiv preprint arXiv:1402.3367 [math-ph], 2014. See Footnote 8. - _N. J. A. Sloane_, Mar 26 2014
• Index entries for algebraic numbers, degree 2.

Crossrefs

Cf. A188934, A246725.

Programs

Mathematica

FromContinuedFraction[{3/2, 3, {3/2, 3}}]
ContinuedFraction[%, 25]  (* [1,1,3,1,1,3,1,1,3,...] *)
RealDigits[N[%%, 120]]  (* A188485 *)
N[%%%, 40]
RealDigits[(3+Sqrt[17])/4,10,120][[1]] (* Harvey P. Dale, May 09 2025 *)