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A380200 a(n) = A379343(A379343(n)).

%I A380200 #15 Mar 19 2025 10:10:34
%S A380200 1,5,2,4,3,6,12,7,14,9,11,8,13,10,15,23,16,25,18,27,20,22,17,24,19,26,
%T A380200 21,28,38,29,40,31,42,33,44,35,37,30,39,32,41,34,43,36,45,57,46,59,48,
%U A380200 61,50,63,52,65,54,56,47,58,49,60,51,62,53,64,55,66
%N A380200 a(n) = A379343(A379343(n)).
%C A380200 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 A380200 The sequence can also be regarded as a table read by upward antidiagonals. For n>1 row n joins two consecutive antidiagonals.
%C A380200 The sequence is an intra-block permutation of the positive integers.
%C A380200 Generalization of Cantor numbering method.
%H A380200 Boris Putievskiy, <a href="/A380200/b380200.txt">Table of n, a(n) for n = 1..9730</a>
%H A380200 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 A380200 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.
%F A380200 Linear sequence: (a(1), a(2), ..., a(A000384(n+1))) is a permutation of the positive integers from 1 to A000384(n+1). (a(1), a(2), ..., a(A000384(n+1))) = (A379343(1), A379343(2), ..., A379343(A000384(n+1)))^2. (a(1), a(2), ..., a(A000384(n+1))) = (A379343(1), A379343(2), ..., A379343(A000384(n+1)))^(-1).
%F A380200 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 + m if k < m and k == 1 (mod 2), P(n,k) = k - 1 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 = 2n - 1.
%e A380200 Triangle array begins:
%e A380200   k=    1  2   3  4   5   6   7   8   9
%e A380200   n=1:  1;
%e A380200   n=2:  5, 2,  4, 3,  6;
%e A380200   n=3: 12, 7, 14, 9, 11,  8, 13, 10, 15;
%e A380200 (1, 5, 2, ..., 10, 15) = (A379343(1), A379343(2), A379343(3), ..., A379343(14), A379343(15))^2.
%e A380200 (1, 5, 2, ..., 10, 15) = (A379343(1), A379343(2), A379343(3), ..., A379343(14), A379343(15))^(-1).
%e A380200 For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
%e A380200    1,  2,  6,  9, 15, ...
%e A380200    5,  3, 14, 10, 27, ...
%e A380200    4,  7, 13, 18, 26, ...
%e A380200   12,  8, 25, 19, 42, ...
%e A380200   11, 16, 24, 31, 41, ...
%e A380200   ...
%e A380200 Subtracting (n-1)*(2*n-3) from each term in row n produces a permutation of numbers from 1 to 4*n-3:
%e A380200   1;
%e A380200   4, 1, 3, 2, 5;
%e A380200   6, 1, 8, 3, 5, 2, 7, 4, 9.
%t A380200 P[n_,k_]:=Module[{m=2*n-1},If[k<m,If[OddQ[k],k+m,k-1],If[OddQ[k],k,k-m+1]]]
%t A380200 Nmax=3;Flatten[Table[P[n,k]+(n-1)*(2*n-3),{n,1,Nmax},{k,1,4*n-3}]]
%Y A380200 Cf. A000384, A016813 (row lengths), A379343.
%K A380200 nonn,tabf
%O A380200 1,2
%A A380200 _Boris Putievskiy_, Jan 16 2025