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.

1, 3, 4, 5, 7, 12, 13, 19, 32, 35, 51, 86, 94, 90, 92, 91, 137, 228, 251, 365, 616, 673, 981, 1654, 1808, 1731, 2635, 4366, 4818, 4592, 4705, 7001, 11706, 12854, 12280, 12567, 18707, 31274, 34344, 32809, 49981, 82790, 91376, 87083, 132771, 219854

Offset: 0

Author

Reed Kelly, Jul 11 2006

Keywords

• easy
• nonn

Comments

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.

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.

Examples

			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.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A069202.

Programs

Mathematica

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]
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 *)

Formula

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).