A189967 Decimal expansion of (7+sqrt(105))/4, which has periodic continued fractions [4,3,4,1,4,3,4,1...] and [7/2, 1, 7/2, 1, ...].

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

Offset

1,1

Comments

Let R denote a rectangle whose shape (i.e., length/width) is (7+sqrt(105))/4. This rectangle can be partitioned into squares in a manner that matches the continued fraction [4,3,4,1,4,3,4,1...]. It can also be partitioned into rectangles of shape 3/2 and 3 so as to match the continued fraction [7/2, 1, 7/2, 1, ...]. For details, see A188635.

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Example

4.311737691489899595805259670130262997684...

Mathematica

FromContinuedFraction[{7/2, 1, {7/2, 1}}]
ContinuedFraction[%, 25]  (* [4, 3, 4, 1, 4, 3, 4, 1...] *)
RealDigits[N[%%, 120]]  (* A189968 *)
N[%%%, 40]

Prog

(PARI) (7+sqrt(105))/4 \\ G. C. Greubel, Jan 12 2018
(Magma) (7+Sqrt(105))/4 // G. C. Greubel, Jan 12 2018

Crossrefs

Cf. A188635, A189967.

Sequence in context: A371809 A157464 A046546 * A139623 A278072 A358710

Adjacent sequences: A189964 A189965 A189966 * A189968 A189969 A189970

Keyword

nonn,cons

Author

Clark Kimberling, May 05 2011

Status

approved