A082844 - OEIS

%I #40 Feb 14 2024 17:27:38

%S 3,2,3,2,2,3,2,3,2,2,3,2,3,2,3,2,2,3,2,3,2,2,3,2,3,2,3,2,2,3,2,3,2,2,

%T 3,2,3,2,2,3,2,3,2,3,2,2,3,2,3,2,2,3,2,3,2,3,2,2,3,2,3,2,2,3,2,3,2,2,

%U 3,2,3,2,3,2,2,3,2,3,2,2,3,2,3,2,3,2,2,3,2,3,2,2,3,2,3,2,3,2,2,3,2,3,2,2,3

%N Start with 3,2 and apply the rule a(a(1)+a(2)+...+a(n)) = a(n), fill in any undefined terms with a(t) = 2 if a(t-1) = 3 and a(t) = 3 if a(t-1) = 2.

%C a(1)=3, a(2)=2, a(a(1)+a(2)+...+a(n)) = a(n) and a(a(1)+a(2)+...+a(n)+1) = 5-a(n).

%C More generally, sequence a(n) = floor(r*(n+2))-floor(r*(n+1)), r = (1/2) *(z+sqrt(z^2+4)), z integer >=1, is defined with a(1), a(2) and a(a(1)+a(2)+...+a(n)+f(z)) = a(n); a(a(1)+a(2)+...+a(n)+f(z)+1) = (2z+1)-a(n) where f(1)=0, f(z)=z-2 for z>=2.

%C Conjecture: a(n) = A097509(n+1). - _Benedict W. J. Irwin_, Mar 13 2016. [See the discussion in A097509. - _N. J. A. Sloane_, Mar 09 2021]

%C Theorem: Referring to the solution to Problem B6 in the 81st William Lowell Putnam Mathematical Competition (see link), in the notation of the first solution, the sequence a(n) = c_{n+1} indexed from 1 equals the present sequence, A082844. - Manjul Bhargava, Kiran Kedlaya, and Lenny Ng, Sep 09 2021.

%H Manjul Bhargava, Kiran Kedlaya, and Lenny Ng, <a href="https://kskedlaya.org/putnam-archive/2020s.pdf">Solutions to the 81st William Lowell Putnam Mathematical Competition</a>

%H Luke Schaeffer, Jeffrey Shallit, and Stefan Zorcic, <a href="https://arxiv.org/abs/2402.08331">Beatty Sequences for a Quadratic Irrational: Decidability and Applications</a>, arXiv:2402.08331 [math.NT], 2024. See pp. 17-19.

%H N. J. A. Sloane, <a href="/A115004/a115004.txt">Families of Essentially Identical Sequences</a>, Mar 24 2021 (Includes this sequence)

%F a(n) = floor(r*(n+2))-floor(r*(n+1)) where r=1+sqrt(2).

%p A082844:=n->floor((1+sqrt(2))*(n+2))-floor((1+sqrt(2))*(n+1)): seq(A082844(n), n=1..100); # _Wesley Ivan Hurt_, Mar 13 2016

%t With[{r=1+Sqrt[2]},Table[Floor[r*(n+2)]-Floor[r*(n+1)],{n,110}]] (* _Harvey P. Dale_, Oct 10 2012 *)

%o (Magma) [Floor((1+Sqrt(2))*(n+2))-Floor((1+Sqrt(2))*(n+1)) : n in [1..100]]; // _Wesley Ivan Hurt_, Mar 13 2016

%Y Cf. A082389, A082845, A097509.

%Y The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A003151 as the parent: A003151, A001951, A001952, A003152, A006337, A080763, A082844 (conjectured), A097509, A159684, A188037, A245219 (conjectured), A276862. - _N. J. A. Sloane_, Mar 09 2021

%K nonn,easy

%O 1,1

%A _Benoit Cloitre_, Apr 15 2003; revised Jun 07 2003