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

1,4

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.

Links

Table of n, a(n) for n=1..78.

Example

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,
  ...

Prog

(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))

Crossrefs

Cf. A316269, A321478, A321479.

Sequence in context: A232628 A087889 A285110 * A192064 A225182 A014710

Adjacent sequences: A323015 A323016 A323017 * A323019 A323020 A323021

Keyword

nonn,tabl

Author

Jianing Song, Jan 07 2019

Status

approved