This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A380817 #45 Jun 24 2025 09:58:37
%S A380817 1,2,3,4,5,6,9,10,7,8,11,12,13,14,15,20,21,18,19,16,17,22,23,24,25,26,
%T A380817 27,28,35,36,33,34,31,32,29,30,37,38,39,40,41,42,43,44,45,54,55,52,53,
%U A380817 50,51,48,49,46,47,56,57,58,59,60,61,62,63,64,65,66
%N A380817 a(n) = A379343(A380245(n)).
%C A380817 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 A380817 The sequence can also be regarded as a table read by upward antidiagonals. For n>1, row n joins two consecutive antidiagonals.
%C A380817 The sequence is a self-inverse permutation of the positive integers.
%C A380817 In particular, the initial {a(1), a(2), ..., a(A000384(n+1))} is self-inverse.
%C A380817 The sequence is an intra-block permutation of the positive integers.
%C A380817 Generalization of the Cantor numbering method.
%C A380817 A379343 and A378684 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, A000027 (the identity permutation), A381662, A376214, A379343, A380200, A378684, A379342, A380245, A381664, A380815, A381663. For subgroups and the Cayley table of the group A4 see Boris Putievskiy (2025) link. - _Boris Putievskiy_, Apr 17 2025
%C A380817 This sequence and A381968 generate via composition a finite non-abelian group of permutations of positive integers, isomorphic to the dihedral group D4. The list of the 8 elements of that group: this sequence, A000027 (the identity permutation), A381968, A381662, A382499, A382679, A376214, A382680. For subgroups and the Cayley table of the group D4 see Putievskiy (D4 (I)) link. - _Boris Putievskiy_, Jun 09 2025
%C A380817 This sequence, A378762 and A381968 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, A382679, A376214, A382680, A378762, 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 A380817 Boris Putievskiy, <a href="/A380817/b380817.txt">Table of n, a(n) for n = 1..9730</a>
%H A380817 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 A380817 Boris Putievskiy, <a href="/A379343/a379343_1.pdf">The Alternating Group A4: Subgroups and the Cayley Table</a> (2025).
%H A380817 Boris Putievskiy, <a href="/A381968/a381968.pdf">The Dihedral Group D4 (I): Subgroups and the Cayley Table</a> (D4 (I)).
%H A380817 Boris Putievskiy, <a href="/A378762/a378762.pdf">The Direct Product D4xC2: Subgroups and the Cayley Table</a>.
%H A380817 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 A380817 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlternatingGroup.html">Alternating Group</a>.
%H A380817 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DihedralGroupD4.html">Dihedral Group D_4</a>.
%H A380817 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.
%F A380817 For 1 <= k <= 4n - 3, T(n,k) = A000384(n-1) + P(n,k), P(n,k) = -k + m - 1 if k < m and k == 1 (mod 2), P(n,k) = -k + m + 1 if k < m and k == 0 (mod 2), P(n,k) = k if k >= m, where m = 2*n - 1.
%e A380817 Triangle array begins:
%e A380817 k= 1 2 3 4 5 6 7 8 9
%e A380817 n=1: 1;
%e A380817 n=2: 2, 3, 4, 5, 6;
%e A380817 n=3: 9, 10, 7, 8, 11, 12, 13, 14, 15;
%e A380817 For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
%e A380817 1, 3, 6, 8, 15, ...
%e A380817 2, 5, 7, 14, 16, ...
%e A380817 4, 10, 13, 19, 26, ...
%e A380817 9, 12, 18, 25, 31, ...
%e A380817 11, 21, 24, 34, 41, ...
%e A380817 ...
%e A380817 Subtracting (n-1)*(2*n-3) from each term in row n produces a permutation of numbers from 1 to 4*n-3:
%e A380817 1,
%e A380817 1, 2, 3, 4, 5;
%e A380817 3, 4, 1, 2, 5, 6, 7, 8, 9.
%e A380817 All permutations are self-inverse.
%t A380817 T[n_,k_]:=(n-1)*(2*n-3)+Module[{m=2*n-1},If[k<m,If[OddQ[k],-k+m-1,-k+m+1],k]]
%t A380817 Nmax=3;Flatten[Table[T[n,k],{n,1,Nmax},{k,1,4*n-3}]]
%Y A380817 Cf. A000027, A000384, A016813 (row lengths), A056023, A376214, A378684, A378762, A379342, A379343, A380200, A380245, A380815, A381662, A381663, A381664, A381968, A382499, A382679, A382680, A383419, A383589, A383590, A383722, A383723, A383724.
%K A380817 nonn,tabf
%O A380817 1,2
%A A380817 _Boris Putievskiy_, Feb 04 2025
%E A380817 Name corrected by _Pontus von Brömssen_, Jun 24 2025