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 A339046 #16 Feb 06 2022 14:59:03
%S A339046 1,1,2,1,4,2,3,1,4,2,3,5,6,1,4,7,2,8,5,1,4,5,9,3,2,8,10,7,6,1,4,3,12,
%T A339046 9,10,2,8,6,11,5,7,1,4,2,8,7,13,11,14,1,4,16,13,2,8,15,9,3,12,14,5,6,
%U A339046 7,11,10,1,4,16,2,8,11,5,20,17,10,19,13,1,4,16,18,3,12,2,8,9,13,6,2,8,9,13,6,1,4,16,18,3,12
%N A339046 Irregular triangle read by rows: row n gives the complete quadrupling system modulo N = 2*n + 1, for n >= 0.
%C A339046 The length of row n is given by phi(2*n + 1), with phi = A000010, for n >= 0.
%C A339046 The quadrupling sequence modulo N = 2*n + 1, for n >= 0, has entries QS(N, s(N,i), j) = s(N,i)*4^j (mod N), with j >= 0, and certain positive integer seeds s(N, i), for i = 1, 2, ..., S(N) = A339049((N-1)/2), where gcd(s(N, i), N) = 1 (restricted seeds modulo N). These quadrupling sequences are periodic with period length P(N) = A053447((N-1)/2) (order of 4 modulo N). Only the periods (cycles) QS(N, s(N,i)) = {QS(N, s(N, i), j)}_{j=0..P(N)-1}, for i = 1, 2, ..., S(N), are listed.
%C A339046 N = 1 (n = 0) is special: one takes here the restricted residue system modulo N not as [0] but as [1]. The order of 4 modulo 1 is 1, because 4^1 == 1 (mod 1) (== 0 (mod 1)).
%C A339046 In order to obtain the complete system of quadrupling sequences one starts with seed s(N, 1) = 1, and if all numbers from the smallest positive reduced residue system modulo N (called RRS(N), given in row N of A038566) are obtained, i.e., if P(N) = #RRS(N) = phi(N) = A000010(N), then the system is complete. Otherwise the smallest missing number from RRS(N) is taken as new seed s(N, 2), etc. until the system is complete. This means that the number of seeds needed is S(N) given above.
%C A339046 This entry generalizes A337712, given together with _Gary W. Adamson_. See also A337936.
%F A339046 T(n, k) gives the k-th entry in the complete quadrupling system modulo N = 2*n + 1, for n >= 0, with the S(N) = A339049((N-1)/2) cycles of length A053447((N-1)/2) written in row n. See the comment above for QS(N,s(N,i)), i = 1, 2, ..., S(N).
%e A339046 The irregular triangle begins (the vertical bar separates the cycles):
%e A339046 n, N \ k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 ...
%e A339046 0, 1: 1
%e A339046 1, 3: 1|2
%e A339046 2, 5: 1 4| 2 3
%e A339046 3, 7: 1 4 2| 3 5 6
%e A339046 4, 9: 1 4 7| 2 8 5
%e A339046 5, 11: 1 4 5 9 3| 2 8 10 7 6
%e A339046 6, 13: 1 4 3 12 9 10| 2 8 6 11 5 7
%e A339046 7, 15: 1 4| 2 8| 7 13|11 14
%e A339046 8, 17: 1 4 16 13| 2 8 15 9| 3 12 14 5| 6 7 11 10
%e A339046 9, 19: 1 4 16 7 9 17 11 6 5| 2 8 13 14 18 15 3 12 10
%e A339046 10, 21: 1 4 16| 2 8 11| 5 20 17|10 19 13
%e A339046 11, 23: 1 4 16 18 3 12 2 8 9 13 6| 2 8 9 13 6 1 4 16 18 3 12
%e A339046 12, 25: 1 4 16 14 6 24 21 9 11 19| 2 8 7 3 12 23 17 18 22 13
%e A339046 13, 27: 1 4 16 10 13 25 19 22 7| 2 8 5 20 26 23 11 17 14
%e A339046 ...
%e A339046 n = 14, N = 29: 1 4 16 6 24 9 7 28 25 13 23 5 20 22 | 2 8 3 12 19 18 14 27 21 26 17 10 11 15,
%e A339046 n = 15, N = 31: 1 4 16 2 8 | 3 12 17 6 24 | 5 20 18 10 9 | 7 28 19 14 25 | 11 13 21 22 26 | 15 29 23 30 27.
%e A339046 ...
%Y A339046 Cf. A000010, A053447, A337712 (doubling), A337936 (tripling), A339049.
%K A339046 nonn,tabf,easy
%O A339046 0,3
%A A339046 _Wolfdieter Lang_, Dec 13 2020