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

Author

Benoit Cloitre, Apr 15 2003; revised Jun 07 2003

Keywords

• nonn
• easy

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

• 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)

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

Programs

Magma

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

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 *)

Python

from math import isqrt
def A082844(n): return 1+isqrt((n+2)**2<<1)-isqrt((n+1)**2<<1) # Chai Wah Wu, May 24 2025

Formula

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