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%I #7 Feb 11 2014 19:05:33
%S 6,22,97,108,114,495,559,2972,3092,3124,3147,3154,3329,3367,3483,3643,
%T 3711,3748,3756,3982,4009,4767,17435,17782,17796,17863,17892,17897,
%U 18079,18139,18422,18580,18644,18688,18784,18804,18952,19739,19868
%N Numbers n such that h(n) = h(n-1) + h(n-2), where h(n) is the length of the sequence {n, f(n), f(f(n)), ...., 1} in the Collatz (or 3x + 1) problem. (The earliest "1" is meant.)
%C Recall that f(n) = n/2 if n is even; = 3n + 1 if n is odd.
%e n, f(n), f(f(n)), ...., 1 for n = 22, 21, 20, respectively, are: 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1; 21, 64, 32, 16, 8, 4, 2, 1; 20, 10, 5, 16, 8, 4, 2, 1. Hence h(22) = 16 = 8 + 8 = h(21) + h(20) and 22 belongs to the sequence.
%t f[n_] := If[EvenQ[n], n/2, 3n+1]; h[n_] := Module[{a, i}, i=n; a=1; While[i>1, a++; i=f[i]]; a]; Select[Range[3, 19900], h[ # ]==h[ #-1]+h[ #-2]&]
%Y Cf. A078419, A078420.
%K nonn
%O 1,1
%A _Joseph L. Pe_, Dec 29 2002
%E Extended by _Robert G. Wilson v_, Dec 30 2002
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