This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A316711 #40 Dec 19 2023 14:21:40
%S A316711 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,
%T A316711 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,
%U A316711 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
%N A316711 Decimal expansion of s:= t/(t - 1), with the tribonacci constant t = A058265.
%C A316711 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.
%C A316711 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
%H A316711 Wolfdieter Lang, <a href="https://arxiv.org/abs/1810.09787">The Tribonacci and ABC Representations of Numbers are Equivalent</a>, arXiv preprint arXiv:1810.09787 [math.NT], 2018.
%H A316711 Wikipedia, <a href="https://en.wikipedia.org/wiki/Beatty_sequence">Beatty Sequence </a>
%F A316711 s = t/(t - 1) with the tribonacci constant t = A058265, the real root of the cubic x^3 - x^2 - x - 1.
%F A316711 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)).
%F A316711 From _Wolfdieter Lang_, Sep 15 2022: (Start)
%F A316711 s = 1 + ((1 + (1/9)*sqrt(33))/4)^(1/3)+(1/3)*((1 + (1/9)*sqrt(33))/4)^(-1/3).
%F A316711 s = 1 + ((1 + (1/9)*sqrt(33))/4)^(1/3) + ((1 - (1/9)*sqrt(33))/4)^(1/3).
%F A316711 s = 1 + (2/3)*sqrt(3)*cosh((1/3)*arccosh((3/4)*sqrt(3))). (End)
%F A316711 From _Dimitri Papadopoulos_, Nov 07 2023: (Start)
%F A316711 s = 1 + t^3/(t^3 - 1) = 1 + A276801/(A276801 - 1).
%F A316711 s = 1 + t^2/(t+1). (End)
%e A316711 s = 2.191487883953118747061354268227517293474691021874288091009781338617685...
%p A316711 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
%t A316711 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 *)
%Y A316711 Cf. A317202, A058265, A158919, A172278, A276801, A357463.
%K A316711 nonn,cons
%O A316711 1,1
%A A316711 _Wolfdieter Lang_, Sep 07 2018