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%I #8 Nov 19 2019 12:46:35
%S 1,3,4,5,7,12,13,19,32,35,51,86,94,90,92,91,137,228,251,365,616,673,
%T 981,1654,1808,1731,2635,4366,4818,4592,4705,7001,11706,12854,12280,
%U 12567,18707,31274,34344,32809,49981,82790,91376,87083,132771,219854
%N 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 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 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 Harvey P. Dale, <a href="/A120424/b120424.txt">Table of n, a(n) for n = 0..1000</a>
%F 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 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 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 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 Cf. A069202.
%K easy,nonn
%O 0,2
%A _Reed Kelly_, Jul 11 2006
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