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A120424 Having specified two initial terms, the "Half-Fibonacci" sequence proceeds like the Fibonacci sequence, except that the terms are halved before being added if they are even.

%I A120424 #8 Nov 19 2019 12:46:35
%S A120424 1,3,4,5,7,12,13,19,32,35,51,86,94,90,92,91,137,228,251,365,616,673,
%T A120424 981,1654,1808,1731,2635,4366,4818,4592,4705,7001,11706,12854,12280,
%U A120424 12567,18707,31274,34344,32809,49981,82790,91376,87083,132771,219854
%N A120424 Having specified two initial terms, the "Half-Fibonacci" sequence proceeds like the Fibonacci sequence, except that the terms are halved before being added if they are even.
%C A120424 For sequences that are infinitely increasing, the following are possible conjectures. Half of the terms are even in the limit. There are infinitely many consecutive pairs that differ by 1.
%C A120424 This is essentially a variant of the Collatz - Fibonacci mixture described in A069202. Instead of conditionally dividing the result by 2, this sequence conditionally divides the two previous terms by 2. The initial two terms of A069202 are 1,2, which corresponds to the initial terms 1,4 for this sequence.
%H A120424 Harvey P. Dale, <a href="/A120424/b120424.txt">Table of n, a(n) for n = 0..1000</a>
%F A120424 a(n) = (a(n-1) if a(n-1) is odd, else a(n-1)/2) + (a(n-2) if a(n-2) is odd, else a(n-2)/2).
%e A120424 Given a(21)=100 and a(22)=117, then a(23)=50+117=167. Given a(13)=64 and a(14)=68, then a(15)=32+34=66.
%t A120424 HalfFib[a_, b_, n_] := Module[{HF, i}, HF = {a, b}; For [i = 3, i < n, i++, HF = Append[HF, HF[[i - 2]]/(2 - Mod[HF[[i - 2]], 2]) + HF[[i - 1]]/(2 - Mod[HF[[i - 1]], 2])]]; HF] HalfFib[1,3,100]
%t A120424 nxt[{a_,b_}]:={b,If[EvenQ[a],a/2,a]+If[EvenQ[b],b/2,b]}; NestList[nxt,{1,3},50][[All,1]] (* _Harvey P. Dale_, Nov 19 2019 *)
%Y A120424 Cf. A069202.
%K A120424 easy,nonn
%O A120424 0,2
%A A120424 _Reed Kelly_, Jul 11 2006