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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.

%I A381705 #27 Apr 02 2025 20:57:50
%S A381705 3,6,13,32,87,250,737,2196,6571,19694,59061,177160,531455,1594338,
%T A381705 4782985,14348924,43046739,129140182,387420509,1162261488,3486784423,
%U A381705 10460353226,31381059633,94143178852,282429536507,847288609470,2541865828357,7625597485016,22876792454991
%N 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.
%C A381705 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.
%C A381705 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.
%C A381705 The peak of a given sequence is 2^(3^(n-1)). See A023365.
%H A381705 David Dewan, <a href="/A381705/a381705.pdf">Unimodal Collatz Sequences</a>
%H A381705 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-7,3).
%F A381705 a(n) = n + 3^(n-1) + 1.
%F A381705 From _Stefano Spezia_, Mar 07 2025: (Start)
%F A381705 G.f.: x*(3 - 9*x + 4*x^2)/((1 - x)^2*(1 - 3*x)).
%F A381705 E.g.f.: (exp(3*x) + 3*exp(x)*(1 + x) - 4)/3. (End)
%e A381705 For n=2, the shortest unimodal sequence has length a(2) = 6 terms and one such sequence is
%e A381705   3 -> 5 ->  8  -> 4 -> 2 -> 1
%e A381705     \-----/     \----------/
%e A381705   2 increases, then decreases
%t A381705 LinearRecurrence[{5,-7,3},{3,6,13},29] (* _James C. McMahon_, Apr 02 2025 *)
%Y A381705 Cf. A014682, A100702, A287319, A221905.
%K A381705 nonn,easy
%O A381705 1,1
%A A381705 _David Dewan_, Mar 04 2025