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

1,5

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.

Links

Table of n, a(n) for n=1..63.

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.

Example

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.

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}]

Crossrefs

Cf. A064414, A230457, A232656.

Sequence in context: A125762 A286216 A280727 * A196302 A307186 A060784

Adjacent sequences: A232354 A232355 A232356 * A232358 A232359 A232360

Keyword

nonn

Author

Brandon Avila and Tanya Khovanova, Nov 22 2013

Status

approved