A232357 The number of pairs of numbers below n that, when generating a Fibonacci-like sequence modulo n, do not contain zero.

0, 0, 0, 0, 4, 0, 0, 24, 0, 16, 20, 48, 84, 0, 36, 120, 144, 144, 36, 64, 288, 80, 0, 360, 104, 336, 0, 288, 448, 144, 60, 504, 580, 864, 196, 912, 684, 792, 756, 760, 880, 1152, 0, 920, 324, 1056, 1472, 1800, 0, 416, 1296, 1344, 1404, 1440, 2504, 2040, 1620, 1792, 116, 1584, 2820, 2040, 2880

Offset: 1

Author

Brandon Avila and Tanya Khovanova, Nov 22 2013

Keywords

• nonn

Comments

a(n) = 0 iff n is in A064414, a(n) is not equal to zero iff n is in A230457.

a(n) + A232656(n) = n^2.

Examples

			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.
Any Fibonacci like sequence contains elements divisible by 2, 3, or 4. Thus, a(2) = a(3) = a(4) = 0.
		

Links

• B. Avila and T. Khovanova, Free Fibonacci Sequences, arXiv preprint arXiv:1403.4614 [math.NT], 2014 and J. Int. Seq. 17 (2014) # 14.8.5.

Crossrefs

Cf. A064414, A230457, A232656.

Programs

Mathematica

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}]