cp's OEIS Frontend

A378684 a(n) = A378200(A378200(n)).

%I A378684 #26 Apr 02 2025 15:07:52
%S A378684 1,3,5,4,2,6,8,14,10,12,11,9,13,7,15,17,27,19,25,21,23,22,20,24,18,26,
%T A378684 16,28,30,44,32,42,34,40,36,38,37,35,39,33,41,31,43,29,45,47,65,49,63,
%U A378684 51,61,53,59,55,57,56,54,58,52,60,50,62,48,64,46,66
%N A378684 a(n) = A378200(A378200(n)).
%C A378684 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 A378684 The sequence can also be regarded as a table read by upward antidiagonals. For n>1 row n joins two consecutive antidiagonals.
%C A378684 The sequence is an intra-block permutation of the positive integers.
%C A378684 Generalization of Cantor numbering method.
%C A378684 The sequence A378200 generates the cyclic group C6 under composition. The elements of C6 are the successive compositions of A378200 with itself: A378684(n) = A378200(A378200(n)) = A378200(n)^2, A378762(n) = A378200(n)^3, A379342(n) = A378200(n)^4, A378705(n) = A378200(n)^5. The identity element is A000027(n) = A378200(n)^6. - _Boris Putievskiy_, Jan 10 2025
%C A378684 This sequence and A379343 generate via composition a finite non-abelian group of permutations of positive integers, isomorphic to the alternating group A4. The list of the 12 elements of that group: this sequence, A379342 (the inverse permutation), A000027 (the identity permutation), A381662, A380817, A376214, A379343, A380200, A380245, A381664, A380815, A381663. For subgroups and the Cayley table of the group A4 see Boris Putievskiy (2025) link. - _Boris Putievskiy_, Mar 28 2025
%H A378684 Boris Putievskiy, <a href="/A378684/b378684.txt">Table of n, a(n) for n = 1..9730</a>
%H A378684 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 A378684 Boris Putievskiy, <a href="/A379343/a379343_1.pdf">The Alternating Group A4: Subgroups and the Cayley Table</a> (2025).
%H A378684 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlternatingGroup.html">Alternating Group</a>.
%H A378684 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.
%F A378684 Linear sequence: (a(1), a(2), ..., a(A000384(n+1))) is permutation of the positive integers from 1 to A000384(n+1). (a(1), a(2), ..., a(A000384(n+1))) = (A378200(1), A378200(2), ..., A378200(A000384(n+1)))^2.
%F A378684 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 + 1 if k < m and k == 1 (mod 2), P(n,k) = 2*m - k if k < m and k == 0 (mod 2), P(n,k) = k if k >= m and k == 1 (mod 2), P(n,k) = 2*m - k - 1 if k >= m and k == 0 (mod 2), where m = 2*n - 1.
%e A378684 Triangle array begins:
%e A378684   k=   1   2   3   4   5   6   7   8   9
%e A378684   n=1: 1;
%e A378684   n=2: 3,  5,  4,  2,  6;
%e A378684   n=3: 8, 14, 10, 12, 11,  9, 13,  7, 15;
%e A378684 (1, 3, 5, ..., 7, 15) = (A378200(1), A378200(2),  A378200(3), ..., A378200(14), A378200(15))^2.
%e A378684 For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
%e A378684    1,  5,  6, 12, 15, ...
%e A378684    3,  2, 10,  7, 21, ...
%e A378684    4, 14, 13, 25, 26, ...
%e A378684    8,  9, 19, 18, 34, ...
%e A378684   11, 27, 24, 42, 41, ...
%e A378684   ...
%e A378684 Subtracting (n-1)*(2*n-3) from each term in row n produces a permutation of numbers from 1 to 4*n-3:
%e A378684   1;
%e A378684   2, 4, 3, 1, 5;
%e A378684   2, 8, 4, 6, 5, 3, 7, 1, 9.
%t A378684 P[n_,k_]:=Module[{m=2*n-1},If[k<m,If[OddQ[k],k+1,2*m-k],If[OddQ[k],k,2*m-k-1]]]
%t A378684 Nmax=3;Flatten[Table[P[n,k]+(n-1)*(2*n-3),{n,1,Nmax},{k,1,4*n-3}]]
%Y A378684 Cf.  A000027, A000384, A016813 (row lengths), A376214,  A379342, A379343, A380200, A380245, A380815, A380817, A381662, A381663, A381664.
%K A378684 nonn,tabf
%O A378684 1,2
%A A378684 _Boris Putievskiy_, Dec 03 2024