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A379342 a(n) = A378684(A378684(n)).

%I A379342 #25 Mar 31 2025 11:58:16
%S A379342 1,5,2,4,3,6,14,7,12,9,11,10,13,8,15,27,16,25,18,23,20,22,21,24,19,26,
%T A379342 17,28,44,29,42,31,40,33,38,35,37,36,39,34,41,32,43,30,45,65,46,63,48,
%U A379342 61,50,59,52,57,54,56,55,58,53,60,51,62,49,64,47,66
%N A379342 a(n) = A378684(A378684(n)).
%C A379342 The 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 A379342 The sequence can also be regarded as a table read by upward antidiagonals. For n>1 row n joins two consecutive antidiagonals.
%C A379342 The sequence is an intra-block permutation of the positive integers.
%C A379342 Generalization of Cantor numbering method.
%C A379342 The sequence A378200 generates the cyclic group C6 under composition. The elements of C6 are the successive compositions of A378200 with itself: A378684(n) = A378200(A378200(n)) = A378200(n)^2, A378762(n) = A378200(n)^3, A379342(n) = A378200(n)^4, A378705(n) = A378200(n)^5. The identity element is A000027(n) = A378200(n)^6. - _Boris Putievskiy_, Jan 03 2025
%C A379342 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, A378684 (the inverse permutation), A000027 (the identity permutation), A381662, A380817, A376214, A379343, A380200, A380245, A381664, A380815, A381663. For subgroups and the Cayley table of the group A4 see Boris Putievskiy (2025) link. - _Boris Putievskiy_, Mar 30 2025
%H A379342 Boris Putievskiy, <a href="/A379342/b379342.txt">Table of n, a(n) for n = 1..9730</a>
%H A379342 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 A379342 Boris Putievskiy, <a href="/A379343/a379343_1.pdf">The Alternating Group A4: Subgroups and the Cayley Table</a> (2025).
%H A379342 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlternatingGroup.html">Alternating Group</
%H A379342 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.
%F A379342 Linear sequence: (a(1), a(2), ..., a(A000384(n+1))) is a permutation of the positive integers from 1 to A000384(n+1). ord(a(1), a(2), ..., a(A000384(n+1))) = 3, where ord is the order of the permutation. (a(1), a(2), ..., a(A000384(n+1))) = (A378684(1), A378684(2), ..., A378684(A000384(n+1)))^(-1).
%F A379342 Triangular array T(n,k) for 1 <= k <= 4n - 3 (see Example): T(n,k) = A000384(n-1) + P(n,k), P(n,k) = max(k, 4n - 3 - k) if k == 1 (mod 2), P(n,k) = min(k - 1, 4n - 2 - k) if k == 0 (mod 2).
%e A379342 Triangle array begins:
%e A379342   k=   1   2   3   4   5   6   7   8   9
%e A379342   n=1: 1;
%e A379342   n=2: 5,  2,  4,  3,  6;
%e A379342   n=3: 14, 7, 12,  9, 11, 10, 13,  8, 15;
%e A379342   ...
%e A379342 (1, 5, 2, ..., 8, 15) = (A378684(1), A378684(2), A378684(3), ..., A378684(14), A378684(15))^2.
%e A379342 (1, 5, 2, ..., 8, 15) = (A378684(1), A378684(2), A378684(3), ..., A378684(14), A378684(15))^(-1).
%e A379342 For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
%e A379342    1,  2,  6,  9, 15, ...
%e A379342    5,  3, 12,  8, 23, ...
%e A379342    4,  7, 13, 18, 26, ...
%e A379342   14, 10, 25, 19, 40, ...
%e A379342   11, 16, 24, 31, 41, ...
%e A379342   ...
%e A379342 Subtracting (n-1)*(2*n-3) from each term in row n produces a permutation of numbers from 1 to 4*n-3:
%e A379342   1;
%e A379342   4, 1, 3, 2, 5;
%e A379342   8, 1, 6, 3, 5, 4, 7, 2, 9.
%t A379342 P[n_,k_]:=If[OddQ[k],Max[k,4 n-3-k],Min[k-1,4 n-2-k]]
%t A379342 Nmax=3;Flatten[Table[P[n,k]+(n-1)*(2*n-3),{n,1,Nmax},{k,1,4*n-3}]]
%Y A379342 Cf. A000027, A000384, A016813 (row lengths), A376214, A378684, A379343, A380200, A380245, A380815, A380817, A381662, A381663, A381664.
%K A379342 nonn,tabf
%O A379342 1,2
%A A379342 _Boris Putievskiy_, Dec 21 2024