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%I #17 Jan 23 2019 09:32:53
%S 1,1,1,1,2,2,1,1,1,1,1,4,4,4,4,1,2,2,1,2,2,1,2,1,2,2,1,2,1,2,1,2,1,2,
%T 1,2,1,2,2,1,2,2,1,2,2,1,4,2,4,2,1,2,4,2,4,1,1,2,2,1,2,2,1,2,2,1,1,2,
%U 2,1,2,2,1,2,2,1,2,2
%N T(n,k) = A321477(n,k)/A321476(n,k), 0 <= k <= n - 1.
%C For Lucas sequences, say, rows in A172236, 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 {A172236(k,m) modulo m} is given as A321477(n,k), and the rank, which is defined as the smallest l > 0 such that n divides A172236(k,l), is given as A321476(n,k). T(n,k) is their ratio, which is the multiplicative order of A172236(k,A321476(n,k)+1) modulo n.
%C T(n,k) has value 1, 2 or 4. This is because A172236(k,m+1)^4 == 1 (mod A172236(k,m)). For n > 2, T(n,k) = 4 iff A321476(n,k) is odd; 1 iff A321476(n,k) is even but not divisible by 4; 2 iff A321476(n,k) is divisible by 4. See A172236 for some further properties.
%C Let p be an odd prime. If p == 3 (mod 4), the p^e-th row consists of only 1 and 2; if p == 5 (mod 8), the p^e-th row consists of only 1 and 4.
%e Table begins
%e 1,
%e 1, 1,
%e 1, 2, 2,
%e 1, 1, 1, 1,
%e 1, 4, 4, 4, 4,
%e 1, 2, 2, 1, 2, 2,
%e 1, 2, 1, 2, 2, 1, 2,
%e 1, 2, 1, 2, 1, 2, 1, 2,
%e 1, 2, 2, 1, 2, 2, 1, 2, 2,
%e 1, 4, 2, 4, 2, 1, 2, 4, 2, 4,
%e ...
%o (PARI) A172236(k, m) = ([k, 1; 1, 0]^m)[2, 1]
%o T(n, k) = my(i=1); while(A172236(k, i)%n!=0, i++); znorder(Mod(A172236(k, i+1), n))
%Y Cf. A172236, A321476, A321477.
%K nonn,tabl
%O 1,5
%A _Jianing Song_, Jan 07 2019
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