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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.

%I A335003 #16 May 28 2025 11:14:29
%S A335003 3,5,10,9,36,18,17,136,68,34,33,528,264,132,66,65,2080,1040,520,260,
%T A335003 130,129,8256,4128,2064,1032,516,258,257,32896,16448,8224,4112,2056,
%U A335003 1028,514,513,131328,65664,32832,16416,8208,4104,2052,1026,1025,524800,262400,131200,65600,32800,16400,8200,4100,2050
%N 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.
%C A335003 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.
%H A335003 Michel Marcus, <a href="/A335003/b335003.txt">Rows n = 1..100 of the triangle, flattened</a>
%H A335003 Hassan Sedaghat, <a href="https://arxiv.org/abs/2004.07357">Bounded Orbits of Quadratic Collatz-type Recursions</a>, arXiv:2004.07357 [math.DS], 2020.
%H A335003 Hassan Sedaghat, <a href="https://arxiv.org/abs/2005.10712">Dual Nature of Orbits of the Divide-or-Choose 2 Rule: A Quadratic Collatz-type Recursion</a>, arXiv:2005.10712 [math.NT], 2020.
%e A335003 Triangle begins:
%e A335003    3;
%e A335003    5, 10;
%e A335003    9, 36, 18;
%e A335003   17, 136, 68, 34;
%e A335003   33, 528, 264, 132, 66;
%e A335003   ...
%t A335003 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 *)
%o A335003 (PARI) f(n) = if (n%2, binomial(n, 2), n/2);
%o A335003 row(n) = my(m=2^n+1, v=vector(n)); v[1] = m; for (i=2, n, v[i] = f(v[i-1])); v;
%Y A335003 Cf. A000051 (1st column), A052548 (right diagonal).
%K A335003 nonn,tabl
%O A335003 1,1
%A A335003 _Michel Marcus_, May 22 2020