Fibonacci-Collatz sequence

[Joseph Biberstine]

The following (pathetic) code suggests no non-constant monotonic
sequences {a(k)} for a(1),a(2) <= 300.  Great sequence by the way.

Clear[x, y, u, monq, s, sd, fib, col, fc];
monq[s_] := (sd = Table[s[[i+1]]-s[[i]], {i,1,Length[s]-1}];
If[Length[Union[NonNegative /@ sd]] == 1 || Length[Union[NonPositive /@
sd]] == 1, 1, 0]);
fc[a_, b_, n_] := fc[a, b, n] = Which[n == 1, a, n == 2, b, True, (fib =
fc[a, b, n - 1] + fc[a, b, n - 2]; If[OddQ[fib], 3*fib + 1, fib/2])];
Table[fc[1, 2, k], {k, 1, 30}]
Select[Table[{a, b, monq[Table[fc[a, b, k], {k, 1, 8}]]}, {a, 1, 300},
{b, 1, 300}], #[[3]] == 1 && #[[1]] != #[[2]] &]

zak seidov wrote:
> Neil, seqfans,
> 
> not yet submitted,
> pending gurus' opinions,
> thanks, Zak
> 
> %I A000001
> %S A000001
> 1,2,10,6,8,7,46,160,103,790,2680,1735,13246,44944,29095,222118,753640,487879,3724558,12637312,8180935,62454742,211907032,137180887,1047263758,3553333936,2300298847,17560898350,59583591592,38572244971
> %N A000001 Fibonacci-Collatz sequence
> %C A000001 Case a(1)=1;a(2)=1 gives "all ones"
> sequence  1,1,1,1,1,1,1 (A000012);
> similarly cases a(1)=a(2)=b give "all b's" sequences
> b,b,b,b,b,b.
> Do all other cases  give in general increasing
> sequences (not-necessarily strictly monotonic)? 
> %F A000001 a(1)=1;a(2)=2; fib=a(n-1)+a(n-2); if fib is
> odd then a(n)=3*fib+1 else a(n)=fib/2
> %t A000001
> a[1]=1;a[2]=2;a[n_]:=a[n]=(fib=a[n-1]+a[n-2];col=If[OddQ[fib],3*fib+1,fib/2]);Table[a[n],{n,30}]
> %Y A000001 A000012
> %O A000001 1
> %K A000001 ,nonn,
> %A A000001 Zak Seidov (zakseidov at yahoo.com), Sep 12
> 2006
> 
> 
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