A376796 - OEIS

%I #30 Jan 03 2025 02:58:30

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

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

%H Boris Putievskiy, <a href="/A376796/b376796.txt">Table of n, a(n) for n = 1..9870</a>

%H 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 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PolygonalNumber.html">Polygonal Number</a>.

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.

%F (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 Table begins:

%e   k =      1   2   3   4   5

%e -----------------------------------

%e   n =  1:  1,  1,  1,  1,  1, ...

%e   n =  2:  4,  3,  5,  6,  2, ...

%e   n =  3:  5,  6,  2,  4,  3, ...

%e   n =  4:  3,  5,  6,  2,  4, ...

%e   n =  5:  6,  2,  4,  3,  5, ...

%e   n =  6:  2,  4,  3,  5,  6, ...

%e Column k = 1 contains the start of A376180.

%e 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 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 Result=P+(L-1)^2*L/2;Result] (*A376180*)

%t composeSequence[a_,n_,k_]:=Nest[a,n,k]

%t Nmax=5;T=Table[composeSequence[a,n,k],{n,1,Nmax},{k,1,Nmax}];T

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

%K nonn,tabl

%O 1,2

%A _Boris Putievskiy_, Dec 22 2024