A323018 T(n,k) = A321479(n,k)/A321478(n,k), 0 <= k <= n - 1.

1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 1

Offset: 1

Author

Jianing Song, Jan 07 2019

Keywords

• nonn
• tabl

Comments

For Lucas sequences, say, rows in A316269, we are mainly concerned about the periods, ranks and the ratios of the periods to the ranks of them modulo a given integer n. The period of {A316269(k,m) modulo m} is given as A321479(n,k), and the rank, which is defined as the smallest l > 0 such that n divides A316269(k,l), is given as A321478(n,k). T(n,k) is their ratio, which is the multiplicative order of A316269(k, A321478(n,k)+1) modulo n.

T(n,k) has value 1 or 2. This is because A316269(k,m+1)^2 == 1 (mod A316269(k,m)). See A172236 for some further properties.

It seems that the n-th row contains more 2's than 1's unless n is a power of 2, in which case the numbers of 1's and 2's are always the same if n >= 4.

Examples

			Table begins
  1,
  1, 1,
  2, 2, 1,
  2, 2, 1, 1,
  2, 2, 1, 2, 1,
  2, 2, 1, 2, 1, 1,
  2, 2, 1, 2, 2, 2, 1,
  2, 2, 1, 2, 1, 2, 1, 1,
  2, 2, 1, 2, 2, 1, 2, 2, 1,
  2, 2, 1, 2, 1, 2, 1, 1, 1, 1,
  ...
		

Crossrefs

Cf. A316269, A321478, A321479.

Programs

PARI

A316269(k, m) = ([k, -1; 1, 0]^m)[2, 1]
T(n, k) = my(i=1); while(A316269(k, i)%n!=0, i++); znorder(Mod(A316269(k, i+1), n))