A385938 a(n) = 2*n/3 if n == 0 (mod 3), (2*n+1)/3 if n == 1 (mod 3), (7*n+1)/3 if n == 2 (mod 3).

0, 1, 5, 2, 3, 12, 4, 5, 19, 6, 7, 26, 8, 9, 33, 10, 11, 40, 12, 13, 47, 14, 15, 54, 16, 17, 61, 18, 19, 68, 20, 21, 75, 22, 23, 82, 24, 25, 89, 26, 27, 96, 28, 29, 103, 30, 31, 110, 32, 33, 117, 34, 35, 124, 36, 37, 131, 38, 39, 138, 40, 41, 145, 42, 43, 152, 44, 45, 159

Offset: 0

Author

Miquel Cerda, Jul 13 2025

Keywords

• nonn
• easy

Comments

Ternary modular function with three cases based on residue modulo 3.

The function defines a dynamical system with multiple periodic attractors.

Fixed point at 1: a(1) = 1.

From Miquel Cerda, Aug 06 2025: (Start)

Every nonnegative integer k appears at least once as a value in the sequence.

Inverse formulas (for possible preimages):

If k is even: one preimage is n = 3*k/2.

If k is odd: one preimage is n = (3*k - 1)/2.

If k == 5 (mod 7): there is an additional preimage: n = 3*(k - 5)/7 + 2. (End)

Examples

			a(0) = 2*0/3 = 0. a(1) = (2*1+1)/3 = 1. a(2) = (7*2+1)/3 = 5. a(3) = 2*3/3 = 2.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Crossrefs

Cf. A385893 (cycle of length 130 in this dynamical system).

Cf. A332057 (near definition).

Programs

Mathematica

a[x_] := Which[Mod[x, 3] == 0, 2*x/3, Mod[x, 3] == 1, (2*x + 1)/3, Mod[x, 3] == 2, (7*x + 1)/3]; Table[a[n], {n, 0, 50}]

PARI

a(n) = if(n%3==0, 2*n/3, if(n%3==1, (2*n+1)/3, (7*n+1)/3))

Python

def A385938(n):
    q, r = divmod(n,3)
    return (q<<1)+r if r<2 else 7*q+5 # Chai Wah Wu, Jul 17 2025

Formula

G.f.: x*(1+5*x+2*x^2+x^3+2*x^4) / ( (x-1)^2*(1+x+x^2)^2 ). - R. J. Mathar, Jul 30 2025