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 A378705 #21 Mar 17 2025 07:00:56
%S A378705 1,3,5,6,2,4,10,12,8,14,15,7,13,9,11,21,23,19,25,17,27,28,16,26,18,24,
%T A378705 20,22,36,38,34,40,32,42,30,44,45,29,43,31,41,33,39,35,37,55,57,53,59,
%U A378705 51,61,49,63,47,65,66,46,64,48,62,50,60,52,58,54,56
%N A378705 Inverse permutation to A378200.
%C A378705 The sequence can be arranged in 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 A378705 The sequence can be represented as a table read by upward antidiagonals. For n>1 row n joins two consecutive antidiagonals.
%C A378705 The sequence is an intra-block permutation of the positive integers.
%C A378705 Generalization of Cantor numbering method.
%C A378705 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 12 2025
%H A378705 Boris Putievskiy, <a href="/A378705/b378705.txt">Table of n, a(n) for n = 1..9730</a>
%H A378705 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 A378705 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.
%F A378705 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)))^(-1).
%F A378705 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 - 1 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 if k >= m and k == 0 (mod 2), where m = 2n - 1.
%e A378705 Triangle array begins:
%e A378705 k= 1 2 3 4 5 6 7 8 9
%e A378705 n=1: 1;
%e A378705 n=2: 3, 5, 6, 2, 4;
%e A378705 n=3: 10, 12, 8, 14, 15, 7, 13, 9, 11;
%e A378705 (1, 3, 5, ..., 9, 11) = (A378200(1), A378200(2), A378200(3), ..., A378200(14), A378200(15))^(-1).
%e A378705 For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
%e A378705 1, 5, 4, 14, 11, ...
%e A378705 3, 2, 8, 9, 17, ...
%e A378705 6, 12, 13, 25, 24, ...
%e A378705 10, 7, 19, 18, 32, ...
%e A378705 15, 23, 26, 40, 41, ...
%e A378705 ...
%e A378705 Subtracting (n-1)*(2*n-3) from each term in row n produces a permutation of numbers from 1 to 4*n-3:
%e A378705 1;
%e A378705 2, 4, 5, 1, 3;
%e A378705 4, 6, 2, 8, 9, 1, 7, 3, 5.
%t A378705 P[n_,k_]:=Module[{m=2*n-1},If[k<m,If[OddQ[k],m-k,m+k-1],If[OddQ[k],3*m-k-1,-m+k]]]
%t A378705 Nmax=3;Flatten[Table[P[n,k]+(n-1)*(2*n-3),{n,1,Nmax},{k,1,4n-3}]]
%Y A378705 Cf. A000027, A000384, A016813 (row lengths), A370655, A373498, A374447, A374494, A374531, A375602, A375725, A378200, A378684, A378762, A379342.
%K A378705 nonn,tabf
%O A378705 1,2
%A A378705 _Boris Putievskiy_, Dec 05 2024