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

0,2

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.

Links

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

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).

Example

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.

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 *)

Crossrefs

Cf. A069202.

Sequence in context: A032890 A092859 A173444 * A139440 A102607 A360688

Adjacent sequences: A120421 A120422 A120423 * A120425 A120426 A120427

Keyword

easy,nonn

Author

Reed Kelly, Jul 11 2006

Status

approved