cp's OEIS Frontend

A378762 a(n) = A378200(A378200(A378200(n))).

%I A378762 #39 Jun 16 2025 17:58:58
%S A378762 1,2,3,6,5,4,9,10,7,8,15,14,13,12,11,20,21,18,19,16,17,28,27,26,25,24,
%T A378762 23,22,35,36,33,34,31,32,29,30,45,44,43,42,41,40,39,38,37,54,55,52,53,
%U A378762 50,51,48,49,46,47,66,65,64,63,62,61,60,59,58,57,56
%N A378762 a(n) = A378200(A378200(A378200(n))).
%C A378762 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 A378762 The sequence can also be regarded as a table read by upward antidiagonals. For n>1 row n joins two consecutive antidiagonals.
%C A378762 The sequence is a self-inverse permutation of natural numbers.
%C A378762 The sequence is an intra-block permutation of the positive integers.
%C A378762 Generalization of Cantor numbering method.
%C A378762 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)) = A378200^2(n), a(n) = A378200^3(n), A379342(n) = A378200^4(n), A378705(n) = A378200^5(n). The identity element is A000027(n) = A378200^6(n). - _Boris Putievskiy_, Jan 15 2025
%C A378762 This sequence, A381968 and A380817 generate via composition a finite non-abelian group of permutations of positive integers, isomorphic to the direct product of the dihedral group D4 and the cyclic group C2. The list of the 16 elements of that group: this sequence, A000027 (the identity permutation), A381968, A381662, A382499, A380817, A382679, A376214, A382680, A383419, A383589, A383590, A056023, A383722, A383723, A383724. For subgroups and the Cayley table of the group D4xC2 see Putievskiy link. - _Boris Putievskiy_, Jun 09 2025
%H A378762 Boris Putievskiy, <a href="/A378762/b378762.txt">Table of n, a(n) for n = 1..9730</a>
%H A378762 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 A378762 Boris Putievskiy, <a href="/A378762/a378762.pdf">The Direct Product D4xC2: Subgroups and the Cayley Table</a>.
%H A378762 Groupprops, <a href="https://groupprops.subwiki.org/wiki/Subgroup_structure_of_direct_product_of_D8_and_Z2">Subgroup structure of direct product of D8 and Z2</a>.
%H A378762 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.
%F A378762 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))) = (A378200(1), A378200(2), ..., A378200(A000384(n+1)))^3. (a(1), a(2), ..., a(A000384(n+1))) = (a(1), a(2), ..., a(A000384(n+1)))^(-1).
%F A378762 Triangular array T(n,k) for 1 <= k <= 4n - 3 (see Example): T(n,k) = A000384(n-1) + P(n,k), P(n,k) = m - k - 1 if k < m and k == 1 (mod 2), P(n, k) = m - k + 1 if k < m and k == 0 (mod 2), P(n, k) = 3m - k - 1 if k >= m, where m = 2n - 1.
%e A378762 Triangle array begins:
%e A378762   k=   1   2   3   4   5   6   7   8   9
%e A378762   n=1: 1;
%e A378762   n=2: 2,  3,  6,  5,  4;
%e A378762   n=3: 9, 10,  7,  8, 15, 14, 13, 12, 11;
%e A378762 (1, 2, 3, ..., 12, 11) = (A378200(1), A378200(2), A378200(3), ..., A378200(14), A378200(15))^3.
%e A378762 (1, 2, 3, ..., 12, 11) = (1, 2, 3, ..., 12, 11)^(-1).
%e A378762 For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
%e A378762    1,  3,  4,  8, 11, ...
%e A378762    2,  5,  7, 12, 16, ...
%e A378762    6, 10, 13, 19, 24, ...
%e A378762    9, 14, 18, 25, 31, ...
%e A378762   15, 21, 26, 34, 41, ...
%e A378762   ...
%e A378762 Subtracting (n-1)*(2*n-3) from each term in row n produces a permutation of numbers from 1 to 4*n-3:
%e A378762   1;
%e A378762   1, 2, 5, 4, 3;
%e A378762   3, 4, 1, 2, 9, 8, 7, 6, 5.
%t A378762 P[n_,k_]:=Module[{m=2*n-1},If[k<m,If[OddQ[k],m-k-1,m+1-k],3*m-1-k]]
%t A378762 Nmax=3; Flatten[Table[P[n,k]+(n-1)*(2*n-3),{n,1,Nmax},{k,1,4*n-3}]]
%Y A378762 Cf. A000027, A000384, A016813 (row lengths), A056023, A376214, A378684, A379342, A379343, A380200, A380245, A380815, A380817, A381662, A381663, A381664, A381968, A382499, A382679, A382680, A383419, A383589, A383590, A383722, A383723, A383724.
%K A378762 nonn,tabf
%O A378762 1,2
%A A378762 _Boris Putievskiy_, Dec 06 2024