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.

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, 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, 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

Offset: 1

Author

Boris Putievskiy, Dec 22 2024

Keywords

• nonn
• tabl

Comments

The sequence A376180 generates infinite cyclic group under composition. The identity element is A000027.

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).

Each column is an intra-block permutation of the positive integers.

Examples

			Table begins:
  k =      1   2   3   4   5
-----------------------------------
  n =  1:  1,  1,  1,  1,  1, ...
  n =  2:  4,  3,  5,  6,  2, ...
  n =  3:  5,  6,  2,  4,  3, ...
  n =  4:  3,  5,  6,  2,  4, ...
  n =  5:  6,  2,  4,  3,  5, ...
  n =  6:  2,  4,  3,  5,  6, ...
Column k = 1 contains the start of A376180.
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.
		

Links

• Boris Putievskiy, Table of n, a(n) for n = 1..9870
• Boris Putievskiy, Integer Sequences: Irregular Arrays and Intra-Block Permutations, arXiv:2310.18466 [math.CO], 2023.
• Eric Weisstein's World of Mathematics, Polygonal Number.
• Index entries for sequences that are permutations of the natural numbers.

Crossrefs

Cf. A000326, A002411, A376180, A378198, A378626.

Programs

Mathematica

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];
Result=P+(L-1)^2*L/2;Result] (*A376180*)
composeSequence[a_,n_,k_]:=Nest[a,n,k]
Nmax=5;T=Table[composeSequence[a,n,k],{n,1,Nmax},{k,1,Nmax}];T

Formula

(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.