A381331 a(1) = a(2) = 1; for n > 2, a(n) = floor((n - 2)*a(n - 1)/a(n - 2)) + GCD(n - 2, a(n - 2)).

1, 1, 2, 5, 8, 7, 5, 5, 8, 13, 15, 12, 9, 21, 31, 27, 14, 9, 11, 31, 54, 35, 16, 11, 16, 35, 55, 41, 21, 15, 21, 57, 85, 48, 19, 15, 28, 70, 93, 52, 24, 22, 38, 74, 84, 51, 30, 28, 44, 79, 88, 56, 33, 34, 55, 89, 144, 91, 39, 25, 38, 96, 155, 102, 42, 28, 44, 105, 160, 104

Offset: 1

Author

Ctibor O. Zizka, Feb 20 2025

Keywords

• nonn
• easy

Comments

At n = 499 the sequence settles down and becomes quasi-periodic with a 6-loop. Empiricaly 3 >= a(n + 1)/a(n) >= 1/3. The system is sensitive to the choice of initial terms [a(1),a(2)]. Only some values of initial terms results in a 6-loop like this sequence, the vast majority of initial terms show a "noisy quasiperiodic" like structures in the plot. Trials made for [a(1), a(2)] from [1, 1] to [100, 100] and for n up to 70000. May it be the sequence converges to a 6-loop for some large enough n, independent on the choice of initial terms ?

Examples

			a(1) = 1
a(2) = 1
a(3) = floor(1*1/1) + GCD(1,1) = 2
a(4) = floor(2*2/1) + GCD(2,1) = 5
a(5) = floor(3*5/2) + GCD(3,2) = 8
and so on.
		

Links

• Ray Chandler, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,-1,1,-1,1,0,0,0,0,0,0,1,-1,1,-1,1,-1).

Crossrefs

Cf. A133058, A145102.

Programs

Mathematica

a[n_] := a[n] = If[n < 3, 1, Floor[(n-2)*a[n-1]/a[n-2]] + GCD[n-2, a[n-2]]]; Array[a, 70] (* Amiram Eldar, Feb 20 2025 *)

Formula

For n >= 499:

if n mod 6 = 0, a(n) = 2*n - 1 + 2*((n/2) mod 2).

if n mod 6 = 1, a(n) = n + 2.

if n mod 6 = 2, a(n) = (n + 2)/2.

if n mod 6 = 3, a(n) = (n - 1)/2.

if n mod 6 = 4, a(n) = n - 2 - (n/2) mod 2.

if n mod 6 = 5, a(n) = 2*n - 6 + 3*((n + 1)/2 mod 2).