A381705 Length of iteration sequence of shortest unimodal Collatz (3x+1)/2 sequence that begins with exactly n increases and ends with continuous decreases until reaching 1.

3, 6, 13, 32, 87, 250, 737, 2196, 6571, 19694, 59061, 177160, 531455, 1594338, 4782985, 14348924, 43046739, 129140182, 387420509, 1162261488, 3486784423, 10460353226, 31381059633, 94143178852, 282429536507, 847288609470, 2541865828357, 7625597485016, 22876792454991

Offset: 1

Author

David Dewan, Mar 04 2025

Keywords

• nonn
• easy

Comments

A unimodal Collatz sequence has one peak because it starts with only odd numbers (which increase) followed by only even numbers (which decrease). It uses the rule odd x -> (3x+1)/2.

A sequence of length a(n) starts with exactly n odd numbers and ends with 3^(n-1) even numbers and the final 1 for a total length of n + 3^(n-1) + 1.

The peak of a given sequence is 2^(3^(n-1)). See A023365.

Examples

			For n=2, the shortest unimodal sequence has length a(2) = 6 terms and one such sequence is
  3 -> 5 ->  8  -> 4 -> 2 -> 1
    \-----/     \----------/
  2 increases, then decreases
		

Links

• David Dewan, Unimodal Collatz Sequences
• Index entries for linear recurrences with constant coefficients, signature (5,-7,3).

Crossrefs

Cf. A014682, A100702, A287319, A221905.

Programs

Mathematica

LinearRecurrence[{5,-7,3},{3,6,13},29] (* James C. McMahon, Apr 02 2025 *)

Formula

a(n) = n + 3^(n-1) + 1.

From Stefano Spezia, Mar 07 2025: (Start)

G.f.: x*(3 - 9*x + 4*x^2)/((1 - x)^2*(1 - 3*x)).

E.g.f.: (exp(3*x) + 3*exp(x)*(1 + x) - 4)/3. (End)