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A383724 a(n) = A378762(A382680(n)).

%I A383724 #8 May 11 2025 21:49:18
%S A383724 1,5,3,6,2,4,12,8,14,10,15,7,13,9,11,23,17,25,19,27,21,28,16,26,18,24,
%T A383724 20,22,38,30,40,32,42,34,44,36,45,29,43,31,41,33,39,35,37,57,47,59,49,
%U A383724 61,51,63,53,65,55,66,46,64,48,62,50,60,52,58,54,56
%N A383724 a(n) = A378762(A382680(n)).
%C A383724 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 A383724 The sequence can also be regarded as a table read by upward antidiagonals. For n > 1 row n joins two consecutive antidiagonals.
%C A383724 The sequence is a self-inverse permutation of the positive integers.
%C A383724 In particular, the initial {a(1), a(2), ..., a(A000384(n+1))} is self-inverse.
%C A383724 The sequence is an intra-block permutation of the positive integers.
%C A383724 Generalization of the Cantor numbering method.
%H A383724 Boris Putievskiy, <a href="/A383724/b383724.txt">Table of n, a(n) for n = 1..9730</a>
%H A383724 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 A383724 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.
%F A383724 T(n,k) for 1 <= k <= 4n - 3: T(n,k) = A000384(n-1) + P(n,k), P(n, k) = m + k if k < m and k == 1 (mod 2), P(n, k) = k if k < m and k == 0 (mod 2), P(n, k) = 3m - 1 - k if k >= m and k == 1 (mod 2), P(n, k) = - m + k if k >= m and k == 0 (mod 2), where m = 2n - 1.
%e A383724 Triangle array begins:
%e A383724   k=     1  2   3   4   5  6   7  8   9
%e A383724   n=1:   1;
%e A383724   n=2:   5, 3,  6,  2,  4;
%e A383724   n=3:  12, 8, 14, 10, 15, 7, 13, 9, 11;
%e A383724 (1, 5, 3, ..., 9, 11) = (1, 2, 3, ..., 12, 11) (1, 5, 3, ..., 9, 15). The first permutation on the right-hand side is from Example A378762 and the second from Example A382680.
%e A383724 (1, 5, 3, ..., 9, 11) = (1, 5, 3, ..., 9, 11)^(-1).
%e A383724 For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
%e A383724    1,  3,  4, 10, 11, ...
%e A383724    5,  2, 14,  9, 27, ...
%e A383724    6,  8, 13, 19, 24, ...
%e A383724   12,  7, 25, 18, 42, ...
%e A383724   15, 17, 26, 32, 41, ...
%e A383724   ...
%e A383724 Subtracting (n-1)*(2*n-3) from each term in row n produces a permutation of numbers from 1 to 4*n-3:
%e A383724   1;
%e A383724   4, 2, 5, 1, 3;
%e A383724   6, 2, 8, 4, 9, 1, 7, 3, 5.
%t A383724 T[n_,k_]:=(n-1)*(2*n-3)+Module[{m=2*n-1},If[k<m,If[OddQ[k],m+k,k],If[OddQ[k],3m-1-k,k-m]]]
%t A383724 Nmax=3; Flatten[Table[T[n,k],{n,1,Nmax},{k,1,4*n-3}]]
%Y A383724 Cf. A000027, A000384, A016813 (row lengths), A376214, A378684, A379342, A379343, A380200, A380245, A380815, A380817, A381662, A381663, A381664, A381968, A382499, A382679, A382680.
%K A383724 nonn,tabf
%O A383724 1,2
%A A383724 _Boris Putievskiy_, May 07 2025