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 A379343 #51 Apr 26 2025 05:31:31
%S A379343 1,3,5,4,2,6,8,12,10,14,11,7,13,9,15,17,23,19,25,21,27,22,16,24,18,26,
%T A379343 20,28,30,38,32,40,34,42,36,44,37,29,39,31,41,33,43,35,45,47,57,49,59,
%U A379343 51,61,53,63,55,65,56,46,58,48,60,50,62,52,64,54,66,68,80,70,82,72,84,74,86,76,88,78,90,79,67,81,69,83,71,85,73,87,75,89,77
%N A379343 Square array read by upward antidiagonals: T(n,k) = (2*(k+n-1)^2 + 2*k - 2*n + 3 + (-2*k - 2*n + 3)*(-1)^n - (-1)^k + (-2*k - 2*n + 1)*(-1)^(k+n))/4.
%C A379343 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 A379343 The sequence can also be regarded as a table read by upward antidiagonals. For n>1 row n joins two consecutive antidiagonals.
%C A379343 The sequence is an intra-block permutation of the positive integers.
%C A379343 Generalization of Cantor numbering method.
%C A379343 This sequence 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, A380200 (the inverse permutation), A000027 (the identity permutation), A381662, A380817, A376214, A378684, A379342, A380245, A381664, A380815, A381663. For subgroups and the Cayley table of the group A4 see Boris Putievskiy (2025) link. - _Boris Putievskiy_, Mar 11 2025
%H A379343 Boris Putievskiy, <a href="/A379343/b379343.txt">Table of n, a(n) for n = 1..9730</a>
%H A379343 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 A379343 Boris Putievskiy, <a href="/A379343/a379343_1.pdf">The Alternating Group A4: Subgroups and the Cayley Table</a> (2025).
%H A379343 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlternatingGroup.html">Alternating Group</a>.
%H A379343 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.
%F A379343 Linear sequence: (a(1),a(2), ... a(A000384(n+1))) is a permutation of the positive integers from 1 to A000384(n+1). ord((a(1),a(2), ... a(A000384(n+1)))) = 3, where ord is the order of the permutation.
%F A379343 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(n) and k mod 2 = 1, P(n,k) = k + m(n) - 1, if k < m(n) and k mod 2 = 0, P(n,k) = k, if k => m(n) and k mod 2 = 1, P(n,k) = k - m(n), if k => m(n) and k mod 2 = 0, where m(n) = 2n-1.
%F A379343 G.f.: x*y*(1 + x^8*y^4 + x*(2 + y) + x^7*y^3*(9 + 8*y) - x^3*y*(8 + 8*y - 3*y^2) - x^2*(1 - 8*y - 9*y^2) + x^6*(y^2 - 4*y^3 - 5*y^4) + x^5*(3*y - 8*y^3 - 2*y^4) + 2*x^4*(1 - 2*y - 5*y^2 + y^4))/((1 - x)^3*(1 + x)^2*(1 - x*y)^3*(1 + x*y)^2). - _Stefano Spezia_, Dec 21 2024
%e A379343 Square array begins:
%e A379343 1, 5, 6, 14, 15, ...
%e A379343 3, 2, 10, 9, 21, ...
%e A379343 4, 12, 13, 25, 26, ...
%e A379343 8, 7, 19, 18, 34, ...
%e A379343 11, 23, 24, 40, 41, ...
%e A379343 ...
%e A379343 The first 5 antidiagonals are:
%e A379343 1;
%e A379343 3, 5;
%e A379343 4, 2, 6;
%e A379343 8, 12, 10, 14;
%e A379343 11, 7, 13, 9, 15;
%e A379343 Triangle array begins:
%e A379343 k= 1 2 3 4 5 6 7 8 9
%e A379343 n=1: 1;
%e A379343 n=2: 3, 5, 4, 2, 6;
%e A379343 n=3: 8, 12, 10, 14, 11, 7, 13, 9, 15;
%e A379343 For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table.
%e A379343 ord(1,3,5... 9,15) = 3.
%e A379343 Subtracting (n-1)*(2*n-3) from each term is row n produces a permutation of numbers from 1 to 4*n-3:
%e A379343 1;
%e A379343 2, 4, 3, 1, 5;
%e A379343 2, 6, 4, 8, 5, 1, 7, 3, 9;
%t A379343 T[n_,k_]:=(2*(k+n-1)^2+2k-2n+3+(-2k-2n+3)*(-1)^n-(-1)^k+(-2k-2n+1)*(-1)^(k+n))/4;Table[T[k,n],{k,1,5},{n,1,5}]
%Y A379343 Cf. A016813 (row lengths), A000384, A000027, A381662, A380817, A376214, A380200, A378684, A379342, A380245, A381664, A380815, A381663.
%K A379343 nonn,tabl,easy
%O A379343 1,2
%A A379343 _Boris Putievskiy_, Dec 21 2024