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 A378198 #9 Dec 03 2024 12:46:07
%S A378198 1,2,1,4,2,1,3,3,2,1,5,4,4,2,1,6,5,3,3,2,1,7,6,5,4,4,2,1,10,7,6,5,3,3,
%T A378198 2,1,13,16,7,6,5,4,4,2,1,16,14,9,7,6,5,3,3,2,1,8,9,17,13,7,6,5,4,4,2,
%U A378198 1,11,10,13,12,14,7,6,5,3,3,2,1,14,8,16,14,11,17,7,6,5,4,4,2,1,17,17,10,9,17,8,12,7,6,5,3,3,2,1,19,12,12,16,13,12,10,11,7
%N A378198 Table T(n, k) read by upward antidiagonals. T(n,1) = A375602(n), T(n,2) = A375602(A375602(n)), T(n,3) = A375602(A375602(A375602(n))) and so on.
%C A378198 Sequence A375602 generates an infinite cyclic group under composition. The identity element is A000027.
%C A378198 Each column is triangle 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 length n(n^2 + 1)/2 = A006003(n).
%C A378198 Each column is an intra-block permutation of the positive integers.
%C A378198 For n > 1, each row combines n consecutive antidiagonals.
%C A378198 Generalization of the Cantor numbering method.
%H A378198 Boris Putievskiy, <a href="/A378198/b378198.txt">Table of n, a(n) for n = 1..9870</a>
%H A378198 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 A378198 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.
%F A378198 (T(1,k),T(2,k), ... T(A002817(n),k)) is permutation of the integers from 1 to A002817(n). (T(1,k),T(2,k), ... T(A002817(n),k)) = (T(1,1),T(2,1), ... T(A002817(n),1))^k.
%e A378198 Table begins:
%e A378198 k = 1 2 3 4 5 6
%e A378198 --------------------------------------
%e A378198 n = 1: 1, 1, 1, 1, 1, 1, ...
%e A378198 n = 2: 2, 2, 2, 2, 2, 2, ...
%e A378198 n = 3: 4, 3, 4, 3, 4, 3, ...
%e A378198 n = 4: 3, 4, 3, 4, 3, 4, ...
%e A378198 n = 5: 5, 5, 5, 5, 5, 5, ...
%e A378198 n = 6: 6, 6, 6, 6, 6, 6, ...
%e A378198 n = 7: 7, 7, 7, 7, 7, 7, ...
%e A378198 n = 8: 10, 16, 9, 13, 14, 17, ...
%e A378198 n = 9: 13, 14, 17, 12, 11, 8, ...
%e A378198 n = 10: 16, 9, 13, 14, 17, 12, ...
%e A378198 ...
%e A378198 Column k = 1 contains the start of A375602.
%e A378198 Ord(T(1,1),T(2,1), ... T(7,1)) = 2, ord(T(1,1),T(2,1), ... T(21,1)) = 18, ord(T(1,1),T(2,1), ... T(55,1)) = 1980, ord(T(1,1),T(2,1), ... T(120,1)) = 51480, where ord is order of permutation.
%e A378198 The first 6 antidiagonals are:
%e A378198 1;
%e A378198 2, 1;
%e A378198 4, 2, 1;
%e A378198 3, 3, 2, 1;
%e A378198 5, 4, 4, 2, 1;
%e A378198 6, 5, 3, 3, 2, 1;
%t A378198 a[n_]:=Module[{L,Ld,Rd,P,Result},L=Ceiling[(Sqrt[4*Sqrt[8*n+1]-3]-1)/2]; Ld=Ceiling[(Sqrt[8*n+1]-1)/2]; Rd=n-(Ld-1)*Ld/2; P=L*Rd+Ld-L*(L+1)/2-Max[Rd-(L^2-L+2)/2,0]*(Max[Rd-(L^2-L+2)/2,0]+1)/2; Result=P+(L-1)*L*(L^2-L+2)/8; Result] (*A375602*) composeSequence[a_,n_,k_]:=Nest[a,n,k]
%t A378198 Nmax=10; Kmax=6; T=Table[composeSequence[a,n,k],{n,1,Nmax},{k,1,Kmax}]
%Y A378198 Cf. A000027, A006003 (row lengths), A002817, A370655, A373498, A374447, A374494, A374531, A375602, A375725.
%K A378198 nonn,tabl
%O A378198 1,2
%A A378198 _Boris Putievskiy_, Nov 19 2024