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%I #19 Feb 28 2018 15:23:47
%S 0,0,0,0,4,0,0,24,0,16,20,48,84,0,36,120,144,144,36,64,288,80,0,360,
%T 104,336,0,288,448,144,60,504,580,864,196,912,684,792,756,760,880,
%U 1152,0,920,324,1056,1472,1800,0,416,1296,1344,1404,1440,2504,2040,1620,1792,116,1584,2820,2040,2880
%N The number of pairs of numbers below n that, when generating a Fibonacci-like sequence modulo n, do not contain zero.
%C a(n) = 0 iff n is in A064414, a(n) is not equal to zero iff n is in A230457.
%C a(n) + A232656(n) = n^2.
%H B. Avila and T. Khovanova, <a href="http://arxiv.org/abs/1403.4614">Free Fibonacci Sequences</a>, arXiv preprint arXiv:1403.4614 [math.NT], 2014 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Avila/avila4.html">J. Int. Seq. 17 (2014) # 14.8.5</a>.
%e The sequence 2,1,3,4,2,1 is the sequence of Lucas numbers modulo 5. Lucas numbers are never divisible by 5. The 4 pairs (2,1), (1,3), (3,4), (4,2) are the only pairs that can generate a sequence modulo 5 that doesn't contain zeros. Thus, a(5) = 4.
%e Any Fibonacci like sequence contains elements divisible by 2, 3, or 4. Thus, a(2) = a(3) = a(4) = 0.
%t fibLike[list_] := Append[list, list[[-1]] + list[[-2]]]; Table[Count[Flatten[Table[Count[Nest[fibLike, {n, m}, k^2]/k, _Integer], {n, k-1}, {m, k-1}]], 0], {k, 70}]
%Y Cf. A064414, A230457, A232656.
%K nonn
%O 1,5
%A _Brandon Avila_ and _Tanya Khovanova_, Nov 22 2013
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