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 A378200 #36 Mar 29 2025 18:09:30
%S A378200 1,5,2,6,3,4,12,9,14,7,15,8,13,10,11,23,20,25,18,27,16,28,17,26,19,24,
%T A378200 21,22,38,35,40,33,42,31,44,29,45,30,43,32,41,34,39,36,37,57,54,59,52,
%U A378200 61,50,63,48,65,46,66,47,64,49,62,51,60,53,58,55,56
%N A378200 Square array read by upward antidiagonals: T(n,k) = ((k + n - 1)^2 + (k - n + 1)*(-1)^n + (1 - k - n)*(-1)^k + (1 - k - n)*(-1)^(k + n) + 2)/2.
%C A378200 The sequence can be arranged in a triangular array, read by rows (blocks). Each row is a permutation of a block of consecutive numbers; the blocks are disjoint and every positive number belongs to some block. Row n has a length of 4n-3 = A016813(n-1), n > 0. For n > 1, each row is a pair consecutive antidiagonals.
%C A378200 The sequence is an intra-block permutation of the positive integers.
%C A378200 Generalization of Cantor numbering method.
%C A378200 This sequence generates the cyclic group C6 under composition. The elements of C6 are the successive compositions of this sequence with itself: A378684(n) = a(a(n)) = a^2(n), A378762(n) = a^3(n), A379342(n) = a^4(n), A378705(n) = a^5(n). The identity element is A000027(n) = a^6(n). - _Boris Putievskiy_, Jan 03 2025
%C A378200 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, A379343 (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 28 2025
%H A378200 Boris Putievskiy, <a href="/A378200/b378200.txt">Table of n, a(n) for n = 1..9870</a>
%H A378200 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 A378200 Boris Putievskiy, <a href="/A379343/a379343_1.pdf">The Alternating Group A4: Subgroups and the Cayley Table</a> (2025).
%H A378200 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlternatingGroup.html">Alternating Group</a>.
%H A378200 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.
%F A378200 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 if k < m and k == 1 (mod 2), P(n,k) = m - k if k < m and k == 0 (mod 2), P(n,k) = 3m - k - 1 if k >= m and k == 1 (mod 2), P(n,k)= - m + k + 1 if k >= m and k == 0 (mod 2), where m = 2n - 1.
%F A378200 G.f.: x*y*(1 + y + y^3 + y^4 - x^3*(2 + y + y^2 - 5*y^3 + y^4) + x*(4 - 3*y + y^2 - y^3 - y^4) - x^2*(1 - 3*y + 9*y^2 + 3*y^3 - 2*y^4) + x^4*(2 + y^2 - 2*y^3 + 3*y^4))/((1 - x)^3*(1 + x)^2*(1 - y)^3*(1 + y)^2). - _Stefano Spezia_, Jan 12 2025
%e A378200 Table begins:
%e A378200 1, 2, 4, 7, 11, ...
%e A378200 5, 3, 14, 10, 27, ...
%e A378200 6, 9, 13, 18, 24, ...
%e A378200 12, 8, 25, 19, 42, ...
%e A378200 15, 20, 26, 33, 41, ...
%e A378200 ...
%e A378200 The first 5 antidiagonals are:
%e A378200 1;
%e A378200 5, 2;
%e A378200 6, 3, 4;
%e A378200 12, 9, 14, 7;
%e A378200 15, 8, 13, 10, 11;
%e A378200 Triangle array begins:
%e A378200 k= 1 2 3 4 5 6 7 8 9
%e A378200 n=1: 1;
%e A378200 n=2: 5, 2, 6, 3, 4;
%e A378200 n=3: 12, 9, 14, 7, 15, 8, 13, 10, 11;
%e A378200 Subtracting (n-1)*(2*n-3) from each term is row n produces a permutation of numbers from 1 to 4*n-3:
%e A378200 1;
%e A378200 4, 1, 5, 2, 3;
%e A378200 6, 3, 8, 1, 9, 2, 7, 4, 5.
%t A378200 T[n_, k_] := ((k + n - 1)^2 + (k - n + 1)*(-1)^n + (1 - k - n)*(-1)^k + (1 - k - n)*(-1)^(k + n) + 2)/2;
%t A378200 Table[T[k, n], {k, 1, 5}, {n, 1, 5}]
%Y A378200 Cf. A000027, A000384, A016813 (row lengths), A370655, A373498, A374447, A374494, A374531, A375602, A375725, A376214, A378684, A378705, A378762, A379342, A379343, A380245, A380815, A380817, A381662, A381663, A381664..
%K A378200 nonn,tabl
%O A378200 1,2
%A A378200 _Boris Putievskiy_, Nov 19 2024