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

0,3

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)

Links

Table of n, a(n) for n=0..68.

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

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

Example

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.

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}]

Prog

(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

Crossrefs

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

Cf. A332057 (near definition).

Sequence in context: A160080 A026247 A354236 * A348779 A226693 A306300

Adjacent sequences: A385935 A385936 A385937 * A385939 A385940 A385941

Keyword

nonn,easy

Author

Miquel Cerda, Jul 13 2025

Status

approved