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%I #12 Nov 18 2018 10:03:28
%S 1,2,3,2,8,8,2,6,4,6,2,20,12,12,20,2,24,8,6,8,24,2,16,6,16,16,6,16,2,
%T 12,8,12,4,12,8,12,2,24,24,6,8,8,6,24,24,2,60,12,12,20,6,20,12,12,60,
%U 2,10,24,8,10,24,24,10,8,24,10,2,24,8,6,8,24,4,24,8,6,8,24
%N Regular triangle read by rows: T(n,k) is the period of {A172236(k,m)} modulo n, 0 <= k <= n - 1.
%C The period of {A172236(k,m)} modulo n is the smallest l such that A172236(k,m) == A172236(k,m+l) (mod n) for every m >= 0. Clearly, T(n,k) is divisible by A321476(n,k). Actually, the ratio is always 1, 2 or 4.
%C Though {A172236(0,m)} is not defined, it can be understood as the sequence 0, 1, 0, 1, ... So the first column of each row (apart from the first one) is always 2.
%C Every row excluding the first term is antisymmetric, that is, T(n,k) = T(n,n-k) for 1 <= k <= n - 1.
%C T(n,k) is the LCM of A321476(n,k) and the multiplicative order of (k + sqrt(k^2 + 4))/2 modulo n, where the multiplicative order of u modulo z is the smallest positive integer l such that (u^l - 1)/z is an algebraic integer.
%F Let p be an odd prime. (i) If ((k^2+4)/p) = 1: if p == 1 (mod 4), then T(p^e,k) is divisible by p^(e-1)*(p - 1), and T(p^e,k) is even; if p == 3 (mod 4), then T(p^e,k) is divisible by p^(e-1)*(p - 1) but not divisible by p^(e-1)*(p - 1)/2. Here (a/p) is the Legendre symbol. (ii) If ((k^2+4)/p) = -1, then T(p^e,k) is divisible by 2*p^(e-1)*(p + 1) but not divisible by p^(e-1)*(p + 1). (iii) If k^2 + 4 is divisible by p, then T(p^e,k) = 4*p^e.
%F For e, k > 0, T(2^e,k) = 3*2^(e-1) for odd k and 2^(e-v(k,2)+1) for even k, where v(k,2) is the 2-adic valuation of k.
%F If gcd(n_1,n_2) = 1, then T(n_1*n_2,k) = lcm(T(n_1,k mod n_1),T(n_2, k mod n_2)).
%F For n > 2, a(n,k)/A321476(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.
%F Let p be an odd prime. (i) If ((k^2+4)/p) = 1: if p == 5 (mod 8), then T(p^e,k)/A321476(p^e,k) != 2; if p == 3 (mod 4), then T(p^e,k)/A321476(p^e,k) = 1. (ii) If ((k^2+4)/p) = -1: if p == 1 (mod 4), then T(p^e,k)/A321476(p^e,k) = 4; if p == 3 (mod 4), then T(p^e,k)/A321476(p^e,k) = 2.
%F T(n,k) <= 6*n.
%e Table begins
%e 1;
%e 2, 3;
%e 2, 8, 8;
%e 2, 6, 4, 6;
%e 2, 20, 12, 12, 20;
%e 2, 24, 8, 6, 8, 24;
%e 2, 16, 6, 16, 16, 6, 16;
%e 2, 12, 8, 12, 4, 12, 8, 12;
%e 2, 24, 24, 6, 8, 8, 6, 24, 24;
%e 2, 60, 12, 12, 20, 6, 20, 12, 12, 60;
%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||(A172236(k, i+1)-1)%n!=0, i++); i
%Y Cf. A172236, A321476 (ranks).
%K nonn,tabl
%O 1,2
%A _Jianing Song_, Nov 11 2018
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