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.
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, 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, 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
Offset: 1
Examples
Table begins: k = 1 2 3 4 5 6 -------------------------------------- n = 1: 1, 1, 1, 1, 1, 1, ... n = 2: 4, 3, 2, 4, 3, 2, ... n = 3: 2, 4, 3, 2, 4, 3, ... n = 4: 3, 2, 4, 3, 2, 4, ... n = 5: 6, 5, 6, 5, 6, 5, ... n = 6: 5, 6, 5, 6, 5, 6, ... n = 7: 12, 11, 9, 8, 10, 7, ... n = 8: 10, 7, 12, 11, 9, 8, ... n = 9: 8, 10, 7, 12, 11, 9, ... n = 10: 7, 12, 11, 9, 8, 10, ... n = 11: 9, 8, 10, 7, 12, 11, ... n = 12: 11, 9, 8, 10, 7, 12, ... n = 13: 15, 14, 13, 15, 14, 13, ... n = 14: 13, 15, 14, 13, 15, 14, ... n = 15: 14, 13, 15, 14, 13, 15, ... 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. The first 6 antidiagonals are: 1; 4, 1; 2, 3, 1; 3, 4, 2, 1; 6, 2, 3, 4, 1; 5, 5, 4, 2, 3, 1;
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.
- Index entries for sequences that are permutations of the natural numbers.
Programs
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Mathematica
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*) composeSequence[a_,n_,k_]:=Nest[a,n,k] Nmax=15;Kmax=6;T=Table[composeSequence[a,n,k],{n,1,Nmax},{k,1,Kmax}]
Comments