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A383589 a(n) = A378762(A381662(n)).

%I A383589 #11 May 06 2025 11:24:05
%S A383589 1,2,3,6,5,4,7,10,9,8,15,12,13,14,11,16,21,18,19,20,17,28,23,26,25,24,
%T A383589 27,22,29,36,31,34,33,32,35,30,45,38,43,40,41,42,39,44,37,46,55,48,53,
%U A383589 50,51,52,49,54,47,66,57,64,59,62,61,60,63,58,65,56
%N A383589 a(n) = A378762(A381662(n)).
%C A383589 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 A383589 The sequence can also be regarded as a table read by upward antidiagonals. For n > 1 row n joins two consecutive antidiagonals.
%C A383589 The sequence is a self-inverse permutation of the positive integers.
%C A383589 In particular, the initial {a(1), a(2), ..., a(A000384(n+1))} is self-inverse.
%C A383589 The sequence is an intra-block permutation of the positive integers.
%C A383589 Generalization of the Cantor numbering method.
%H A383589 Boris Putievskiy, <a href="/A383589/b383589.txt">Table of n, a(n) for n = 1..9730</a>
%H A383589 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 A383589 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.
%F A383589 T(n,k) for 1 <= k <= 4n - 3: T(n,k) = A000384(n-1) + P(n,k), P(n, k) = k if k < m and k == 1 (mod 2), P(n, k) = m + 1 - 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) = k if k >= m and k == 0 (mod 2), where m = 2n - 1.
%e A383589 Triangle array begins:
%e A383589   k=    1   2  3  4   5   6   7   8   9
%e A383589   n=1:  1;
%e A383589   n=2:  2,  3, 6, 5,  4;
%e A383589   n=3:  7, 10, 9, 8, 15, 12, 13, 14, 11;
%e A383589 (1, 2, 3, ..., 14, 11) = (1, 2, 3, ..., 12, 11) (1, 2, 3, ..., 12, 15). The first permutation on the right-hand side is from Example A378762 and the second from Example A381662.
%e A383589 (1, 2, 3, ..., 14, 11) = (1, 2, 3, ..., 14, 11)^(-1).
%e A383589 For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
%e A383589    1,  3,  4,  8, 11, ...
%e A383589    2,  5,  9, 14, 20, ...
%e A383589    6, 10, 13, 19, 24, ...
%e A383589    7, 12, 18, 25, 33, ...
%e A383589   15, 21, 26, 34, 41, ...
%e A383589   ...
%e A383589 Subtracting (n-1)*(2*n-3) from each term in row n produces a permutation of numbers from 1 to 4*n-3:
%e A383589   1;
%e A383589   1, 2, 5, 4, 3;
%e A383589   1, 4, 3, 2, 9, 6, 7, 8, 5.
%t A383589 T[n_,k_]:=(n-1)*(2*n-3)+Module[{m=2*n-1},If[k<m,If[OddQ[k],k,m+1-k],If[OddQ[k],3m-1-k,k]]]
%t A383589 Nmax=3; Flatten[Table[T[n,k],{n,1,Nmax},{k,1,4*n-3}]]
%Y A383589 Cf.  A000027, A000384, A016813 (row lengths), A376214, A378684, A379342, A379343, A380200, A380245, A380815, A380817, A381662, A381663, A381664, A381968, A382499, A382679, A382680.
%K A383589 nonn,tabf
%O A383589 1,2
%A A383589 _Boris Putievskiy_, May 01 2025