A335003 Triangle read by rows where the n-th row is the cycle trajectory of 2^n+1 in the divide-or-choose 2 rule.

3, 5, 10, 9, 36, 18, 17, 136, 68, 34, 33, 528, 264, 132, 66, 65, 2080, 1040, 520, 260, 130, 129, 8256, 4128, 2064, 1032, 516, 258, 257, 32896, 16448, 8224, 4112, 2056, 1028, 514, 513, 131328, 65664, 32832, 16416, 8208, 4104, 2052, 1026, 1025, 524800, 262400, 131200, 65600, 32800, 16400, 8200, 4100, 2050

Offset: 1

Author

Michel Marcus, May 22 2020

Keywords

• nonn
• tabl

Comments

The divide-or-choose-2 rule is a quadratic Collatz-type recursion where the map is defined with f(n) = n/2 if n is even, and f(n) = binomial(n, 2) if n is odd.

Examples

			Triangle begins:
   3;
   5, 10;
   9, 36, 18;
  17, 136, 68, 34;
  33, 528, 264, 132, 66;
  ...
		

Links

• Michel Marcus, Rows n = 1..100 of the triangle, flattened
• Hassan Sedaghat, Bounded Orbits of Quadratic Collatz-type Recursions, arXiv:2004.07357 [math.DS], 2020.
• Hassan Sedaghat, Dual Nature of Orbits of the Divide-or-Choose 2 Rule: A Quadratic Collatz-type Recursion, arXiv:2005.10712 [math.NT], 2020.

Crossrefs

Cf. A000051 (1st column), A052548 (right diagonal).

Programs

Mathematica

f[n_] := If[EvenQ[n], n/2, Binomial[n, 2]]; row[n_] := NestWhileList[f, n, f[#] != n &]; Join @@ Table[row[2^n + 1], {n, 1, 10}] (* Amiram Eldar, May 22 2020 *)

PARI

f(n) = if (n%2, binomial(n, 2), n/2);
row(n) = my(m=2^n+1, v=vector(n)); v[1] = m; for (i=2, n, v[i] = f(v[i-1])); v;