A178255 Decimal expansion of (3+sqrt(17))/2.

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

Offset: 1

Author

Klaus Brockhaus, May 24 2010

Keywords

• cons
• nonn

Comments

Continued fraction expansion of (3+sqrt(17))/2 is A109007.

a(n) = A082486(n) for n > 1.

The rectangle R whose shape (i.e., length/width) is (3+sqrt(17))/2 can be partitioned into rectangles of shapes 3 and 3/2 in a manner that matches the periodic continued fraction [3, 3/2, 3, 3/2, ...]. R can also be partitioned into squares so as to match the periodic continued fraction [3, 1, 1, 3, 1, 1,...]. For details, see A188635. - Clark Kimberling, May 07 2011

The positive eigenvalue of the real symmetric 2 X 2 matrix M defined by M(i,j) = max(i,j) = [(1 2), (2 2)] is (3+sqrt(17))/2, while the negative one is (3-sqrt(17))/2. For a generalization, see A085984. - Bernard Schott, Apr 13 2020

A quadratic integer with minimal polynomial x^2 - 3x - 2. - Charles R Greathouse IV, Apr 14 2020

The positive root of x^2 - 3^x - 2. The negative root is -(-3 + sqrt(17))/2 = -0.56155... - Wolfdieter Lang, Dec 10 2022

Examples

			(3+sqrt(17))/2 = 3.56155281280883027491...
		

Links

• Index entries for algebraic numbers, degree 2

Crossrefs

Cf. A082486 (decimal expansion of (5+sqrt(17))/2), A010473 (decimal expansion of sqrt(17)), A109007 (repeat 3, 1, 1), A085984.

Programs

Mathematica

FromContinuedFraction[{3, 3/2, {3, 3/2}}]
ContinuedFraction[%, 100] (* [3,1,1,3,1,1,...] *)
RealDigits[N[%%, 120]]    (* A178255 *)
N[%%%, 40]
(* Clark Kimberling, May 07 2011 *)

PARI

(3+sqrt(17))/2 \\ Charles R Greathouse IV, Apr 14 2020