This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A321478 #14 Nov 18 2018 10:03:37
%S A321478 1,2,3,2,3,3,2,3,4,3,2,3,5,5,3,2,3,6,6,6,3,2,3,7,4,4,7,3,2,3,8,3,4,3,
%T A321478 8,3,2,3,9,6,9,9,6,9,3,2,3,10,15,6,6,6,15,10,3,2,3,11,5,5,6,6,5,5,11,
%U A321478 3,2,3,12,6,6,3,4,3,6,6,12,3
%N A321478 Regular triangle read by rows: T(n,k) is the rank of {A316269(k,m)} modulo n, 0 <= k <= n - 1.
%C A321478 The rank of {A316269(k,m)} modulo n is the smallest l such that n divides A316269(k,l).
%C A321478 Though {A316269(0,m)} is not defined, it can be understood as the sequence 0, 1, 0, -1, 0, 1, 0, -1, ... So the first column of each row (apart from the first one) is always 2.
%C A321478 Though {A316269(1,m)} is not defined, it can be understood as the sequence 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, ... So the second column of each row is always 3.
%C A321478 Every row excluding the first term is antisymmetric, that is, T(n,k) = T(n,n-k) for 1 <= k <= n - 1.
%C A321478 T(n,k) is the multiplicative order of ((k + sqrt(k^2 - 4))/2)^2 modulo n*sqrt(k^2 - 4), 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 A321478 Let p be a prime >= 5. (i) If k^2 - 4 is not divisible by p, then T(p^e,k) is divisible by p^(e-1)*(p - ((k^2-4)/p))/2. Here (a/p) is the Legendre symbol. (ii) If k^2 - 4 is divisible by p, then T(p^e,k) = p^e.
%F A321478 For e >= 2 and 1 < k < 2^e - 1, T(2^e,k) = 3*2^(e-v(k^2-1,2)) 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 A321478 For e > 0 and k > 1, T(3^e,k) = 2*3^(e-v(k,3)) for k divisible by 3 and 3^(e-v(k^2-1,3)+1) otherwise.
%F A321478 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 A321478 T(n,k) <= (3/2)*n.
%e A321478 Table begins
%e A321478 1;
%e A321478 2, 3;
%e A321478 2, 3, 3;
%e A321478 2, 3, 4, 3;
%e A321478 2, 3, 5, 5, 3;
%e A321478 2, 3, 6, 6, 6, 3;
%e A321478 2, 3, 7, 4, 4, 7, 3;
%e A321478 2, 3, 8, 3, 4, 3, 8, 3;
%e A321478 2, 3, 9, 6, 9, 9, 6, 9, 3;
%e A321478 2, 3, 10, 15, 6, 6, 6, 15, 10, 3;
%e A321478 ...
%o A321478 (PARI) A316269(k, m) = ([k, -1; 1, 0]^m)[2, 1]
%o A321478 T(n, k) = my(i=1); while(A316269(k, i)%n!=0, i++); i
%Y A321478 Cf. A316269, A321479 (periods).
%K A321478 nonn,tabl
%O A321478 1,2
%A A321478 _Jianing Song_, Nov 11 2018