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A381663 a(n) = A379342(A380200(n)).

%I A381663 #25 Jun 24 2025 09:58:44
%S A381663 1,3,5,4,2,6,10,14,8,12,11,7,13,9,15,21,27,19,25,17,23,22,16,24,18,26,
%T A381663 20,28,36,44,34,42,32,40,30,38,37,29,39,31,41,33,43,35,45,55,65,53,63,
%U A381663 51,61,49,59,47,57,56,46,58,48,60,50,62,52,64,54,66
%N A381663 a(n) = A379342(A380200(n)).
%C A381663 This sequence can be regarded as a triangular array read by rows. Each row is a permutation of a block of consecutive numbers; the blocks are disjoint and every positive number belongs to some block. The length of row n is 4n-3 = A016813(n-1), n > 0.
%C A381663 The sequence can also be regarded as a table read by upward antidiagonals. For n>1 row n joins two consecutive antidiagonals.
%C A381663 The sequence is an intra-block permutation of the positive integers.
%C A381663 Generalization of the Cantor numbering method.
%C A381663 A379343 and A378684 generate via composition a finite non-abelian group of permutations of positive integers, isomorphic to the alternating group A4. The list of the 12 elements of that group: this sequence, A380815 (the inverse permutation), A000027 (the identity permutation), A381662, A380817, A376214, A379343, A380200, A378684, A379342, A380245, A381664. For subgroups and the Cayley table of the group A4 see Boris Putievskiy (2025) link. - _Boris Putievskiy_, Apr 10 2025
%H A381663 Boris Putievskiy, <a href="/A381663/b381663.txt">Table of n, a(n) for n = 1..9730</a>
%H A381663 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 A381663 Boris Putievskiy, <a href="/A379343/a379343_1.pdf">The Alternating Group A4: Subgroups and the Cayley Table</a> (2025).
%H A381663 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlternatingGroup.html">Alternating Group</a>.
%H A381663 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.
%F A381663 ord(a(1), a(2), ..., a(A000384(n+1))) = 3, where ord is the order of the permutation.
%F A381663 For 1 <= k <= 4n - 3, T(n,k) = A000384(n-1) + P(n,k), P(n,k) = -k + m if k < m and k == 1 (mod 2), P(n,k) = -k + 2m if k < m and k == 0 (mod 2), P(n,k) = k if k >= m and k == 1 (mod 2), P(n,k) = k - m if k >= m and k == 0 (mod 2), where m = 2n - 1.
%e A381663 Triangle array begins:
%e A381663   k=    1   2  3   4   5  6   7  8   9
%e A381663   n=1:  1;
%e A381663   n=2:  3,  5, 4,  2,  6;
%e A381663   n=3: 10, 14, 8, 12, 11, 7, 13, 9, 15;
%e A381663 ord(1, 3, 5, ..., 9, 15) = 3.
%e A381663 For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
%e A381663    1,  5,  6, 12, 15, ...
%e A381663    3,  2,  8,  9, 17, ...
%e A381663    4, 14, 13, 25, 26, ...
%e A381663   10,  7, 19, 18, 32, ...
%e A381663   11, 27, 24, 42, 41, ...
%e A381663   ...
%e A381663 Subtracting (n-1)*(2*n-3) from each term in row n produces a permutation of numbers from 1 to 4*n-3:
%e A381663   1;
%e A381663   2, 4, 3, 1, 5;
%e A381663   4, 8, 2, 6, 5, 1, 7, 3, 9.
%t A381663 T[n_,k_]:=(n-1)*(2*n-3)+Module[{m=2*n-1},If[k<m,If[OddQ[k],m-k,2*m-k],If[OddQ[k],k,k-m]]]
%t A381663 Nmax=3;Flatten[Table[T[n,k],{n,1,Nmax},{k,1,4*n-3}]]
%Y A381663 Cf. A000027, A000384, A016813 (row lengths), A376214, A378684, A379342, A379343, A380200, A380245, A380815, A380817, A381662, A381664.
%K A381663 nonn,tabf
%O A381663 1,2
%A A381663 _Boris Putievskiy_, Mar 03 2025
%E A381663 Name corrected by _Pontus von Brömssen_, Jun 24 2025