A190285 Decimal expansion of (3+sqrt(9+4r))/2, where r=sqrt(3).

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

Offset

1,1

Comments

The rectangle R whose shape (i.e., length/width) is (3+sqrt(9+4r))/2, where r=sqrt(3), can be partitioned into rectangles of shapes 3 and r in a manner that matches the periodic continued fraction [3, r, 3, r, ...]. R can also be partitioned into squares so as to match the nonperiodic continued fraction [3,2,55,6,1,1,1,9,1,1,1,7,2,...] at A190286. For details, see A188635.

Links

Table of n, a(n) for n=1..120.

Index entries for algebraic numbers, degree 4.

Formula

Minimal polynomial: x^4 - 6*x^3 + 9*x^2 - 3. - Amiram Eldar, May 28 2026

Example

3.495507656604924503772866679054481005188610884047732453142114506972152814621....

Mathematica

r=3^(1/2)
FromContinuedFraction[{3, r, {3, r}}]
FullSimplify[%]
RealDigits[N[%%, 120]]
N[%%%, 40]
(* Alternative: *)
RealDigits[(3+Sqrt[9+4Sqrt[3]])/2, 10, 120][[1]] (* Harvey P. Dale, Oct 19 2021 *)

Prog

(PARI) (3+sqrt(9+4*sqrt(3)))/2 \\ Charles R Greathouse IV, May 18 2026

Crossrefs

Cf. A188635, A190286.

Sequence in context: A083437 A011292 A021745 * A276521 A247567 A271529

Adjacent sequences: A190282 A190283 A190284 * A190286 A190287 A190288

Keyword

nonn,cons

Author

Clark Kimberling, May 07 2011

Status

approved