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 A378626 #8 Dec 03 2024 12:46:32
%S A378626 1,4,1,2,3,1,3,4,2,1,6,2,3,4,1,5,5,4,2,3,1,12,6,6,3,4,2,1,10,11,5,5,2,
%T A378626 3,4,1,8,7,9,6,6,4,2,3,1,7,10,12,8,5,5,3,4,2,1,9,12,7,11,10,6,6,2,3,4,
%U A378626 1,11,8,11,12,9,7,5,5,4,2,3,1,15,9,10,9,11,8,12,6,6,3,4,2,1,13,14,8,7,8,9,10,11,5,5,2,3,4,1,14,15,13,10,12,10,8,7,9,6,6,4,2
%N A378626 Table T(n, k) read by upward antidiagonals. T(n,1) = A377137(n), T(n,2) = A377137(A377137(n)), T(n,3) = A377137(A377137(A377137(n))) and so on.
%C A378626 The sequence A377137 generates infinite cyclic group under composition. The identity element is A000027.
%C A378626 Each column is 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.
%C A378626 Row n has length A064455(n). The sequence A064455 is non-monotonic.
%C A378626 The array consists of two triangular arrays alternating row by row.
%C A378626 For odd n, row n consists of permutations of the integers from A001844((n-1)/2) to A265225(n-1). For even n, row n consists of permutations of the integers from A130883(n/2) to A265225(n-1).
%C A378626 Each column is an intra-block permutation of the positive integers.
%H A378626 Boris Putievskiy, <a href="/A378626/b378626.txt">Table of n, a(n) for n = 1..9870</a>
%H A378626 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 A378626 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.
%F A378626 (T(1,k),T(2,k), ... T(A265225(n),k)) is permutation of the integers from 1 to A265225(n). (T(1,k),T(2,k), ... T(A265225(n),k)) = (T(1,1),T(2,1), ... T(A265225(n),1))^k.
%e A378626 Table begins:
%e A378626 k = 1 2 3 4 5 6
%e A378626 --------------------------------------
%e A378626 n = 1: 1, 1, 1, 1, 1, 1, ...
%e A378626 n = 2: 4, 3, 2, 4, 3, 2, ...
%e A378626 n = 3: 2, 4, 3, 2, 4, 3, ...
%e A378626 n = 4: 3, 2, 4, 3, 2, 4, ...
%e A378626 n = 5: 6, 5, 6, 5, 6, 5, ...
%e A378626 n = 6: 5, 6, 5, 6, 5, 6, ...
%e A378626 n = 7: 12, 11, 9, 8, 10, 7, ...
%e A378626 n = 8: 10, 7, 12, 11, 9, 8, ...
%e A378626 n = 9: 8, 10, 7, 12, 11, 9, ...
%e A378626 n = 10: 7, 12, 11, 9, 8, 10, ...
%e A378626 n = 11: 9, 8, 10, 7, 12, 11, ...
%e A378626 n = 12: 11, 9, 8, 10, 7, 12, ...
%e A378626 n = 13: 15, 14, 13, 15, 14, 13, ...
%e A378626 n = 14: 13, 15, 14, 13, 15, 14, ...
%e A378626 n = 15: 14, 13, 15, 14, 13, 15, ...
%e A378626 Column k = 1 contains the start of A377137. Ord(T(1,1),T(2,1), ... T(15,1)) = 6, ord(T(1,1),T(2,1), ... T(24,1)) = 18, ord(T(1,1),T(2,1), ... T(45,1)) = 90, ord(T(1,1),T(2,1), ... T(112,1)) = 1260, where ord is order of permutation.
%e A378626 The first 6 antidiagonals are:
%e A378626 1;
%e A378626 4, 1;
%e A378626 2, 3, 1;
%e A378626 3, 4, 2, 1;
%e A378626 6, 2, 3, 4, 1;
%e A378626 5, 5, 4, 2, 3, 1;
%t A378626 a[n_]:=Module[{L,R,P,Result},L=Ceiling[Max[x/.NSolve[x*(2*(x-1)-Cos[Pi*(x-1)]+5)-4*n==0,x,Reals]]];R=n-If[EvenQ[L],(L^2-L)/2,(L^2-1)/2]; P[(L+1)*(2*L-(-1)^L+5)/4]=If[EvenQ[L],3L/2,(L+1)/2];P[3]=2;P=Abs[2*R-If[EvenQ[L],3L/2,(L+1)/2]-If[2*R<=If[EvenQ[L],3L/2,(L+1)/2]+1,2,1]];Res=P+If[EvenQ[L],(L^2-L)/2,(L^2-1)/2];Result=Res;Result] (*A377137*)
%t A378626 composeSequence[a_,n_,k_]:=Nest[a,n,k]
%t A378626 Nmax=15;Kmax=6;T=Table[composeSequence[a,n,k],{n,1,Nmax},{k,1,Kmax}]
%Y A378626 Cf. A000027, A064455 (row lengths), A265225, A377137, A378127.
%K A378626 nonn,tabl
%O A378626 1,2
%A A378626 _Boris Putievskiy_, Dec 02 2024