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 A376796 #30 Jan 03 2025 02:58:30
%S A376796 1,4,1,5,3,1,3,6,5,1,6,5,2,6,1,2,2,6,4,2,1,13,4,4,2,3,4,1,12,16,3,3,4,
%T A376796 5,3,1,14,10,8,5,5,3,6,5,1,11,9,11,12,6,6,5,2,6,1,15,15,14,15,10,2,2,
%U A376796 6,4,2,1,10,17,17,9,17,11,4,4,2,3,4,1,16,11,18,18,14,18,15,3,3,4,5,3,1,9,8,15,7,7,9,7,17,5,5,3,6,5,1,17,14,12,17,13,13,14,13
%N A376796 Table T(n, k) read by upward antidiagonals. T(n,1) = A376180, T(n,2) = A376180(A376180(n)), T(n,3) = A376180(A376180(A376180(n))) and so on.
%C A376796 The sequence A376180 generates infinite cyclic group under composition. The identity element is A000027.
%C A376796 Each column 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 the row number n in each column is the n-th pentagonal number, A000326(n+1).
%C A376796 Each column is an intra-block permutation of the positive integers.
%H A376796 Boris Putievskiy, <a href="/A376796/b376796.txt">Table of n, a(n) for n = 1..9870</a>
%H A376796 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 A376796 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PolygonalNumber.html">Polygonal Number</a>.
%H A376796 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.
%F A376796 (T(1,k),T(2,k), ... T(A002411(n),k)) is a permutation of the integers from 1 to A002411(n). (T(1,k),T(2,k), ... T(A002411(n),k)) = (T(1,1),T(2,1), ... T(A002411(n),1))^k.
%e A376796 Table begins:
%e A376796 k = 1 2 3 4 5
%e A376796 -----------------------------------
%e A376796 n = 1: 1, 1, 1, 1, 1, ...
%e A376796 n = 2: 4, 3, 5, 6, 2, ...
%e A376796 n = 3: 5, 6, 2, 4, 3, ...
%e A376796 n = 4: 3, 5, 6, 2, 4, ...
%e A376796 n = 5: 6, 2, 4, 3, 5, ...
%e A376796 n = 6: 2, 4, 3, 5, 6, ...
%e A376796 Column k = 1 contains the start of A376180.
%e A376796 Ord(T(1,1),T(2,1), ... T(6,1)) = 5, ord(T(1,1),T(2,1), ... T(18,1)) = 10, ord(T(1,1),T(2,1), ... T(40,1)) = 60, ord(T(1,1),T(2,1), ... T(75,1)) = 420, ord(T(1,1),T(2,1), ... T(126,1)) = 7140, where ord is the order of the permutation.
%t A376796 a[n_]:=Module[{L,R,P,Result},L=Ceiling[Max[x/.NSolve[x^2 (x+1)-2 n==0,x,Reals]]];R=n-((L-1)^2)*L/2;P=Which[OddQ[R]&&OddQ[L*(3*L-1)/2],(L*(3*L-1)/2-R+1+1)/2,OddQ[R]&&EvenQ[L*(3*L-1)/2],(R+L*(3*L-1)/2+1)/2,EvenQ[R]&&OddQ[L*(3*L-1)/2],Ceiling[(L*(3*L-1)/2+1)/2]+R/2,EvenQ[R]&&EvenQ[L*(3*L-1)/2],Ceiling[(L*(3*L-1)/2+1)/2]-R/2];
%t A376796 Result=P+(L-1)^2*L/2;Result] (*A376180*)
%t A376796 composeSequence[a_,n_,k_]:=Nest[a,n,k]
%t A376796 Nmax=5;T=Table[composeSequence[a,n,k],{n,1,Nmax},{k,1,Nmax}];T
%Y A376796 Cf. A000326, A002411, A376180, A378198, A378626.
%K A376796 nonn,tabl
%O A376796 1,2
%A A376796 _Boris Putievskiy_, Dec 22 2024