A317202 -id:A317202 - cp's OEIS frontend

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

Wolfdieter Lang, Sep 07 2018

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

			s = 2.191487883953118747061354268227517293474691021874288091009781338617685...

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

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

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

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 *)

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)

cp's OEIS Frontend

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

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