A190290 Decimal expansion of (3+sqrt(21))/3.

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

Offset: 1

Author

Clark Kimberling, May 07 2011

Keywords

• nonn
• cons

Comments

The rectangle R whose shape (i.e., length/width) is (3+sqrt(21))/3, can be partitioned into rectangles of shapes 3/2 and 2 in a manner that matches the periodic continued fraction [2, 3/2, 2, 3/2, ...]. R can also be partitioned into squares so as to match the periodic continued fraction [2,1,1,8,1,1,2,1,1,8,1,1,2,,...]. For details, see A188635.

Examples

			2.527525231651946668862682397909336162995...
		

Crossrefs

Cf. A188635, A190289.

Programs

Mathematica

FromContinuedFraction[{2, 3/2, {2, 3/2}}]
ContinuedFraction[%, 100]  (* [2,1,1,8,1,1,2,... *)
RealDigits[N[%%, 120]]     (* A190290 *)
N[%%%, 40]

Formula

Equals 1 + Sum_{k>=0} binomial(2*k,k)/7^k. - Amiram Eldar, Aug 03 2020