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A380245 a(n) = A378684(A379343(n)).

%I A380245 #27 Jun 24 2025 09:58:33
%S A380245 1,5,2,4,3,6,14,9,12,7,11,8,13,10,15,27,20,25,18,23,16,22,17,24,19,26,
%T A380245 21,28,44,35,42,33,40,31,38,29,37,30,39,32,41,34,43,36,45,65,54,63,52,
%U A380245 61,50,59,48,57,46,56,47,58,49,60,51,62,53,64,55,66
%N A380245 a(n) = A378684(A379343(n)).
%C A380245 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 A380245 The sequence can also be regarded as a table read by upward antidiagonals. For n>1 row n joins two consecutive antidiagonals.
%C A380245 The sequence is an intra-block permutation of the positive integers.
%C A380245 Generalization of Cantor numbering method.
%C A380245 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, A381664 (the inverse permutation), A000027 (the identity permutation), A381662, A380817, A376214, A379343, A380200, A379342, A378684, A380815, A381663. For subgroups and the Cayley table of the group A4 see Boris Putievskiy (2025) link. - _Boris Putievskiy_, Mar 31 2025
%H A380245 Boris Putievskiy, <a href="/A380245/b380245.txt">Table of n, a(n) for n = 1..9730</a>
%H A380245 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 A380245 Boris Putievskiy, <a href="/A379343/a379343_1.pdf">The Alternating Group A4: Subgroups and the Cayley Table</a> (2025).
%H A380245 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlternatingGroup.html">Alternating Group</a>
%H A380245 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.
%F A380245 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.
%F A380245 Triangular array T(n,k) for 1 <= k <= 4n - 3 (see Example): T(n,k) = A000384(n-1) + P(n,k), P(n,k) = -k + 2*m - 1 if k < m and k == 1 (mod 2), P(n,k) = m - k 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 + 1 if k >= m and k == 0 (mod 2), where m = 2*n - 1.
%e A380245 Triangle array begins:
%e A380245   k=    1  2   3  4   5   6   7   8   9
%e A380245   n=1:  1;
%e A380245   n=2:  5, 2,  4, 3,  6;
%e A380245   n=3: 14, 9, 12, 7, 11,  8, 13, 10, 15;
%e A380245 ord(1, 5, 2, ..., 10, 15) = 3.
%e A380245 For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
%e A380245    1,  2,  6,  7, 15, ...
%e A380245    5,  3, 12, 10, 23, ...
%e A380245    4,  9, 13, 18, 26, ...
%e A380245   14,  8, 25, 19, 40, ...
%e A380245   11, 20, 24, 33, 41, ...
%e A380245   ...
%e A380245 Subtracting (n-1)*(2*n-3) from each term in row n produces a permutation of numbers from 1 to 4*n-3:
%e A380245   1;
%e A380245   4, 1, 3, 2, 5;
%e A380245   8, 3, 6, 1, 5, 2, 7, 4, 9.
%t A380245 P[n_,k_]:=Module[{m=2*n-1},If[k<m,If[OddQ[k],-k+2m-1,m-k],If[OddQ[k],k,k-m+1]]]
%t A380245 Nmax=3;Flatten[Table[P[n,k]+(n-1)*(2*n-3),{n,1,Nmax},{k,1,4*n-3}]]
%Y A380245 Cf. A000027, A000384, A016813 (row lengths), A376214, A378684, A379342, A379343, A380200, A380815, A380817, A381662, A381663, A381664.
%K A380245 nonn,tabf
%O A380245 1,2
%A A380245 _Boris Putievskiy_, Jan 17 2025
%E A380245 Name corrected by _Pontus von Brömssen_, Jun 24 2025