A356033 Decimal expansion of (-1 + sqrt(13))/6 = A223139/3.

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

Offset: 0

Author

Wolfdieter Lang, Aug 29 2022

Keywords

• nonn
• cons
• easy

Comments

This constant r, an algebraic integer of the quadratic number field Q(13), is the positive root of its monic minimal polynomial x^2 + x/3 - 1/3. The negative root is -(1 + sqrt(13))/6 = -A209927/3 = -(A188943 - 1).

r^n = A052533(-n) + A006130(-(n+1))*r, for n >= 0, with A052533(-n) = 3*sqrt(-3)^(-n-2)*Snx(-n-2,1/sqrt(-3)), and A006130(-(n+1)) = sqrt(-3)^(-(n+1))*Snx(-(n+1), 1/sqrt(-3)), with the S-Chebyshev polynomials (see A049310), with S(-n, x) = -S(n-2, x), for n>=2, and S(-1, x) = 0. - Wolfdieter Lang, Nov 27 2023

Examples

			0.4342585459106648821865368779117493243752160956408743687850755...
		

Crossrefs

Cf. A006130, A049310, A052533, A188943, A209927, A223139.

Programs

Mathematica

First[RealDigits[x/.N[Last[Solve[3x^2+x-1==0,x]],78]]] (* Stefano Spezia, Aug 29 2022 *)

Formula

r = (-1 + sqrt(13))/6 = A223139/3 = 1/A209927.