A082844 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.

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, 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, 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

Offset

1,1

Comments

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).

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.

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]

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.

Links

Table of n, a(n) for n=1..105.

Manjul Bhargava, Kiran Kedlaya, and Lenny Ng, Solutions to the 81st William Lowell Putnam Mathematical Competition

Luke Schaeffer, Jeffrey Shallit, and Stefan Zorcic, Beatty Sequences for a Quadratic Irrational: Decidability and Applications, arXiv:2402.08331 [math.NT], 2024. See pp. 17-19.

N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)

Formula

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

Maple

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

Mathematica

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

Prog

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

Crossrefs

Cf. A082389, A082845, A097509.

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

Sequence in context: A064654 A162229 A056564 * A279124 A101406 A363445

Adjacent sequences: A082841 A082842 A082843 * A082845 A082846 A082847

Keyword

nonn,easy

Author

Benoit Cloitre, Apr 15 2003; revised Jun 07 2003

Status

approved