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 A373498 #24 Aug 03 2024 11:10:11
%S A373498 2,1,3,5,9,7,6,8,4,10,12,18,14,20,16,15,17,13,19,11,21,23,31,25,33,27,
%T A373498 35,29,28,30,26,32,24,34,22,36,38,48,40,50,42,52,44,54,46,45,47,43,49,
%U A373498 41,51,39,53,37,55
%N A373498 a(a(a(n))) = A370655(n).
%C A373498 Triangle read by rows where row n is a block of length 4*n-1 which is a permutation of the numbers of its constituents.
%C A373498 Generalization of the Cantor numbering method for two adjacent diagonals. A pair of neighboring diagonals are combined into one block.
%C A373498 The sequence is an intra-block permutation of positive integers.
%C A373498 The sequence A373498 generates the cyclic group C6 under composition. The elements of C6 are the successive compositions of A373498 with itself: A374494 = A373498(A373498) = A373498^2, A370655 = A373498^3, A374531 = A373498^4, A374447 = A373498^5. The identity element is A000027 = A373498^6. - _Boris Putievskiy_, Aug 02 2024
%H A373498 Boris Putievskiy, <a href="/A373498/b373498.txt">Table of n, a(n) for n = 1..9870</a> _Boris Putievskiy_, Aug 02 2024
%H A373498 Boris Putievskiy, <a href="https://arxiv.org/abs/2310.18466">Integer Sequences: Irregular Arrays and Intra-Block Permutations</a>, arXiv:2310.18466 [math.CO], 2023.
%H A373498 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%H A373498 _Boris Putievskiy_, Aug 02 2024
%F A373498 Linear sequence:
%F A373498 a(n) = P(n) + (L(n)-1)*(2*L(n)-1), where L(n) = ceiling((sqrt(8*n+1)-1)/4),
%F A373498 L(n) = A204164(n), R(n) = n - (L(n)-1)*(2*L(n)-1),
%F A373498 P(n) = R(n) + 1 if R(n) <= 2*L(n)-1 and R(n) mod 2 = 1, P(n) = 2*L(n) + R(n) if R(n) <= 2*L(n)-1 and R(n) mod 2 = 0, P(n) = R(n) if R(n) > 2*L(n)-1 and R(n) mod 2 = 1, P(n) = 4*L(n) - R(n) - 1 if R(n) > 2*L(n)-1 and R(n) mod 2 = 0.
%F A373498 Triangular array T(n,k) for 1 <= k <= 4*n-1 (see Example):
%F A373498 T(n,k) = (n-1)*(2*n-1) + P(n,k), where P(n,k) = k + 1 if k <= 2*n-1 and k mod 2 = 1, P(n,k) = 2*n + k if k <= 2*n-1 and k mod 2 = 0,
%F A373498 P(n,k) = k if k > 2*n-1 and k mod 2 = 1, P(n,k) = 4*n - k - 1 if k > 2*n-1 and k mod 2 = 0.
%e A373498 Triangle begins:
%e A373498 k = 1 2 3 4 5 6 7 8 9 10 11
%e A373498 n=1: 2, 1, 3;
%e A373498 n=2: 5, 9, 7, 6, 8, 4, 10;
%e A373498 n=3: 12, 18, 14, 20, 16, 15, 17, 13, 19, 11, 21;
%e A373498 The triangle's rows can be arranged as two successive upward antidiagonals in an array:
%e A373498 2, 3, 7, 10, 16, 21, ...
%e A373498 1, 9, 4, 20, 11, 35, ...
%e A373498 5, 8, 14, 19, 27, 34, ...
%e A373498 6, 18, 13, 33, 24, 52, ...
%e A373498 12, 17, 25, 32, 42, 51, ...
%e A373498 15, 31, 26, 50, 41, 73, ...
%e A373498 Subtracting (n-1)*(2*n-1) from each term in row n is a permutation of 1 .. 4*n-1:
%e A373498 2,1,3,
%e A373498 2,6,4,3,5,1,7,
%e A373498 2,8,4,10,6,5,7,3,9,1,11,
%e A373498 ...
%e A373498 The 3rd power of each permutation is equal to the corresponding permutation in example A370655:
%e A373498 (2,1,3)^3 = (2,1,3),
%e A373498 (2,6,4,3,5,1,7)^3 = (1,2,4,3,5,6,7),
%e A373498 (2,8,4,10,6,5,7,3,9,1,11)^3 = (3,4,1,2,6,5,7,10,9,8,11).
%t A373498 Nmax=21;
%t A373498 a[n_]:=Module[{L,R,P,Result},L=Ceiling[(Sqrt[8*n+1]-1)/4];
%t A373498 R=n-(L-1)*(2*L-1);
%t A373498 P=Which[R<=2*L-1&&Mod[R,2]==1,R+1,R<=2*L-1&&Mod[R,2]==0,R+2*L,R>2*L-1&&Mod[R,2]==1,R,R>2*L-1&&Mod[R,2]==0,4*L-1-R];
%t A373498 Result=P+(L-1)*(2*L-1);
%t A373498 Result]
%t A373498 Table[a[n],{n,1,Nmax}]
%t A373498 Table[a[a[a[n]]],{n,1,Nmax}] (* A370655 *)
%Y A373498 Cf. A000027, A004767 (row lengths), A204164, A370655, A374447, A374494, A374531.
%K A373498 nonn,tabf
%O A373498 1,1
%A A373498 _Boris Putievskiy_, Jun 17 2024