A368120 Repeated application of the Syracuse map over F_2[x] starting from 1+x^3, m = 1+x^2. Represented as the integers resulting from evaluating the polynomial at 2 over Z.

9, 22, 11, 19, 47, 73, 182, 91, 155, 379, 587, 1459, 2495, 6049, 9362, 4681, 11702, 5851, 9947, 24283, 37595, 93403, 159707, 387163, 599195, 1497979, 2546507, 6216499, 9624575, 23910913, 40884482, 20442241

Offset

-2,1

Comments

Shown to be unbounded by K. R. Matthews and G. M. Leigh. Matthews and Leigh start the sequence at 1+x+x^3, which evaluates to 11. See Example 2 of their paper.

1+x^3 is the first starting polynomial in lexicographical order whose repeated Syracuse map is unbounded. For this reason, I start the sequence at 9. Because Matthews and Leigh start the sequence at 11, there is an offset of -2.

Links

Table of n, a(n) for n=-2..29.

K. R. Matthews and G. M. Leigh, A generalization of the Syracuse algorithm in Fq[x], Journal of Number Theory, Volume 25, Issue 3, March 1987, pp. 274-278.

Formula

Equivalent to the following recurrence relation over integers:

a(-2) = 9

a(n+1) =

(a(n) >> 1) XOR (a(n) << 1) for a(n) odd

a(n) >> 1 for a(n) even

where >> and << are bitwise shifts and XOR is bitwise XOR, n >= -2.

a(5*(2^n-1)) = (10*2^(3*2^n)-3)/7, n >= 0 (offset -2).

Example

1 + x^3 = 9
multiply by 1 + x^2, add 1, and divide by x:
x + x^2 + x^4 = 22
divide by x
1 + x + x^3 = 11

Prog

(JavaScript)
let total = '';
let polynomial = 9;
let num_iterate = 32;
for (let i = 0; i < num_iterate; i++) {
  total += polynomial + ', '
if (polynomial % 2 == 0) polynomial >>= 1;
else polynomial = (polynomial >> 1) ^ (polynomial << 1);
}
console.log(total);

Crossrefs

Sequence in context: A342409 A373747 A329007 * A250729 A251292 A177458

Adjacent sequences: A368117 A368118 A368119 * A368121 A368122 A368123

Keyword

nonn,easy

Author

Keith J. Bauer, Dec 13 2023

Status

approved