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A383723 a(n) = A378762(A376214(n)).

%I A383723 #8 May 11 2025 21:57:03
%S A383723 1,2,3,6,5,4,9,8,7,10,15,12,13,14,11,20,17,18,19,16,21,28,23,26,25,24,
%T A383723 27,22,35,30,33,32,31,34,29,36,45,38,43,40,41,42,39,44,37,54,47,52,49,
%U A383723 50,51,48,53,46,55,66,57,64,59,62,61,60,63,58,65,56
%N A383723 a(n) = A378762(A376214(n)).
%C A383723 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 A383723 The sequence can also be regarded as a table read by upward antidiagonals. For n > 1 row n joins two consecutive antidiagonals.
%C A383723 The sequence is a self-inverse permutation of the positive integers.
%C A383723 In particular, the initial {a(1), a(2), ..., a(A000384(n+1))} is self-inverse.
%C A383723 The sequence is an intra-block permutation of the positive integers.
%C A383723 Generalization of the Cantor numbering method.
%H A383723 Boris Putievskiy, <a href="/A383723/b383723.txt">Table of n, a(n) for n = 1..9730</a>
%H A383723 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 A383723 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.
%F A383723 T(n,k) for 1 <= k <= 4n - 3: T(n,k) = A000384(n-1) + P(n,k), P(n, k) = m - 1 - 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) = k if k >= m and k == 0 (mod 2), where m = 2n - 1.
%e A383723 Triangle array begins:
%e A383723   k=    1  2  3   4   5   6   7   8   9
%e A383723   n=1:  1;
%e A383723   n=2:  2, 3, 6,  5,  4;
%e A383723   n=3:  9, 8, 7, 10, 15, 12, 13, 14, 11;
%e A383723 (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 A376214.
%e A383723 (1, 2, 3, ..., 14, 11) = (1, 2, 3, ..., 14, 11)^(-1).
%e A383723 For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
%e A383723    1,  3,  4, 10, 11, ...
%e A383723    2,  5,  7, 14, 16, ...
%e A383723    6,  8, 13, 19, 24, ...
%e A383723    9, 12, 18, 25, 31, ...
%e A383723   15, 17, 26, 32, 41, ...
%e A383723   ...
%e A383723 Subtracting (n-1)*(2*n-3) from each term in row n produces a permutation of numbers from 1 to 4*n-3:
%e A383723   1;
%e A383723   1, 2, 5, 4, 3;
%e A383723   3, 2, 1, 4, 9, 6, 7, 8, 5.
%t A383723 T[n_,k_]:=(n-1)*(2*n-3)+Module[{m=2*n-1},If[k<m,If[OddQ[k],m-1-k,k],If[OddQ[k],3m-1-k,k]]]
%t A383723 Nmax=3; Flatten[Table[T[n,k],{n,1,Nmax},{k,1,4*n-3}]]
%Y A383723 Cf. A000027, A000384, A016813 (row lengths), A376214, A378684, A379342, A379343, A380200, A380245, A380815, A380817, A381662, A381663, A381664, A381968, A382499, A382679, A382680.
%K A383723 nonn,tabf
%O A383723 1,2
%A A383723 _Boris Putievskiy_, May 07 2025