A316711 Decimal expansion of s:= t/(t - 1), with the tribonacci constant t = A058265.

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

Offset: 1

Author

Wolfdieter Lang, Sep 07 2018

Keywords

• nonn
• cons

Comments

Because the tribonacci constant t = A058265 > 1, with Beatty sequence At(n) := floor(n*t), n >= 1 (with At(0) = 0) given in A158919, has the companion sequence Bt := floor(n*s), n >= 1, (with Bt(0) = 0), with 1/t + 1/s = 1, and At and Bt are complementary, disjoint sequences for the positive integers. Note that Bt is not A172278. The first entries n = 0..161 coincide. A172278(162) = 354 but At(193) = A158919(193) = 354, hence A172278 is not complementary together with At. In fact, Bt(162) = 355, which is not a member of At.

s-1 = 1/(t-1) equals the real root of 2*x^3 - 2*x - 1. See the formulas below. - Wolfdieter Lang, Sep 15 2022

Examples

			s = 2.191487883953118747061354268227517293474691021874288091009781338617685...
		

Links

• Wolfdieter Lang, The Tribonacci and ABC Representations of Numbers are Equivalent, arXiv preprint arXiv:1810.09787 [math.NT], 2018.
• Wikipedia, Beatty Sequence

Crossrefs

Cf. A317202, A058265, A158919, A172278, A276801, A357463.

Programs

Maple

Digits := 120: a := (1/4 + sqrt(33)/36)^(1/3): 1 + a + 1/(3*a): evalf(%)*10^98: ListTools:-Reverse(convert(floor(%), base, 10)); # Peter Luschny, Sep 15 2022

Mathematica

With[{t=x/.Solve[x^3-x^2-x-1==0,x][[1]]},RealDigits[t/(t-1),10,120][[1]]] (* Harvey P. Dale, Sep 12 2021 *)

Formula

s = t/(t - 1) with the tribonacci constant t = A058265, the real root of the cubic x^3 - x^2 - x - 1.

s = (1 + (19 + 3*sqrt(33))^(1/3) + (19 - 3*sqrt(33))^(1/3)) / (-2 + (19 + 3*sqrt(33))^(1/3) + (19 - 3*sqrt(33))^(1/3)).

From Wolfdieter Lang, Sep 15 2022: (Start)

s = 1 + ((1 + (1/9)*sqrt(33))/4)^(1/3)+(1/3)*((1 + (1/9)*sqrt(33))/4)^(-1/3).

s = 1 + ((1 + (1/9)*sqrt(33))/4)^(1/3) + ((1 - (1/9)*sqrt(33))/4)^(1/3).

s = 1 + (2/3)*sqrt(3)*cosh((1/3)*arccosh((3/4)*sqrt(3))). (End)

From Dimitri Papadopoulos, Nov 07 2023: (Start)

s = 1 + t^3/(t^3 - 1) = 1 + A276801/(A276801 - 1).

s = 1 + t^2/(t+1). (End)