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A381968 a(a(n)) = A381662(n).

%I A381968 #40 Jun 15 2025 18:15:24
%S A381968 1,5,3,4,2,6,14,8,12,10,11,7,13,9,15,27,17,25,19,23,21,22,16,24,18,26,
%T A381968 20,28,44,30,42,32,40,34,38,36,37,29,39,31,41,33,43,35,45,65,47,63,49,
%U A381968 61,51,59,53,57,55,56,46,58,48,60,50,62,52,64,54,66
%N A381968 a(a(n)) = A381662(n).
%C A381968 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 A381968 The sequence can also be regarded as a table read by upward antidiagonals. For n > 1 row n joins two consecutive antidiagonals.
%C A381968 The sequence is an intra-block permutation of the positive integers.
%C A381968 Generalization of the Cantor numbering method.
%C A381968 This sequence and A380817 generate via composition a finite non-abelian group of permutations of positive integers, isomorphic to the dihedral group D4. The list of the 8 elements of that group: this sequence, A382499 (the inverse permutation), A000027 (the identity permutation), A381662, A380817, A382679, A376214, A382680. For subgroups and the Cayley table of the group D4 see Putievskiy (D4 (I)) link. - _Boris Putievskiy_, Jun 09 2025
%C A381968 This sequence,  A378762 and A380817 generate via composition a finite non-abelian group of permutations of positive integers, isomorphic to the direct product of the dihedral group D4 and the cyclic group C2. The list of the 16 elements of that group: this sequence, A000027 (the identity permutation), A382499 (the inverse permutation), A381662, A380817, A382679, A376214, A382680, A378762, A383419, A383589, A383590, A056023, A383722, A383723, A383724. For subgroups and the Cayley table of the group D4xC2 see Putievskiy link. - _Boris Putievskiy_, Jun 09 2025
%H A381968 Boris Putievskiy, <a href="/A381968/b381968.txt">Table of n, a(n) for n = 1..9730</a>
%H A381968 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 A381968 Boris Putievskiy, <a href="/A381968/a381968.pdf">The Dihedral Group D4 (I): Subgroups and the Cayley Table</a> (D4 (I)).
%H A381968 Boris Putievskiy, <a href="/A378762/a378762.pdf">The Direct Product D4xC2: Subgroups and the Cayley Table</a>.
%H A381968 Groupprops, <a href="https://groupprops.subwiki.org/wiki/Subgroup_structure_of_direct_product_of_D8_and_Z2">Subgroup structure of direct product of D8 and Z2</a>.
%H A381968 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DihedralGroupD4.html">Dihedral Group D_4</a>.
%H A381968 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.
%F A381968 ord(a(1), a(2), ..., a(A000384(n+1))) = 4, where ord is the order of the permutation.
%F A381968 T(n,k) for 1 <= k <= 4n - 3: T(n,k) = A000384(n-1) + P(n,k), P(n,k) = - k + 2m - 1 if k < m and k == 1 (mod 2), P(n,k) = 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 if k >= m and k == 0 (mod 2), where m = 2n - 1.
%e A381968 Triangle array begins:
%e A381968   k=     1  2   3   4   5   6   7   8   9
%e A381968   n=1:   1;
%e A381968   n=2:   5, 3,  4,  2,  6;
%e A381968   n=3:  14, 8, 12, 10, 11,  7, 13,  9, 15;
%e A381968 (1,5,3,...,9,15) (1,5,3,...,9,15) = (1,2,3,...,12,15). The permutation on the right-hand side is from Example A381662.
%e A381968 ord(1,5,3,...,9,15) = 4.
%e A381968 For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
%e A381968    1,  3,  6, 10, 15, ...
%e A381968    5,  2, 12,  9, 23, ...
%e A381968    4,  8, 13, 19, 26, ...
%e A381968   14,  7, 25, 18, 40, ...
%e A381968   11, 17, 24, 32, 41, ...
%e A381968   ...
%e A381968 Subtracting (n-1)*(2*n-3) from each term in row n produces a permutation of numbers from 1 to 4*n-3:
%e A381968   1;
%e A381968   4, 2, 3, 1, 5;
%e A381968   8, 2, 6, 4, 5, 1, 7, 3, 9.
%t A381968 T[n_,k_]:=(n-1)*(2*n-3)+Module[{m=2*n-1},If[k<m,If[OddQ[k],2m-1-k,k],If[OddQ[k],k,k-m]]]
%t A381968 Nmax=3; Flatten[Table[T[n,k],{n,1,Nmax},{k,1,4*n-3}]]
%Y A381968 Cf. A000027, A000384, A016813 (row lengths), A056023, A376214, A378684, A378762, A379342, A379343, A380200, A380245, A380815, A380817, A381662, A381663, A381664, A382499, A382679, A382680, A383419, A383589, A383590, A383722, A383723, A383724.
%K A381968 nonn,tabf
%O A381968 1,2
%A A381968 _Boris Putievskiy_, Mar 12 2025