OFFSET
1,2
COMMENTS
Row n consists of permutation of the integers from F(n+2) to F(n+3) - 1, where F(n) = A000045(n). The permutation is generated using Kevin Ryde's formula from A194959.
The sequence is an intra-block permutation of the positive integers.
LINKS
Boris Putievskiy, Table of n, a(n) for n = 1..6764
Boris Putievskiy, Integer Sequences: Irregular Arrays and Intra-Block Permutations, arXiv:2310.18466 [math.CO], 2023.
FORMULA
T(n,k) for 1 <= k <= F(n) (see Example):
T(n,k) = P(n,k) + F(n+1)-1, T(n,k) = P(n,k) + A000045(n+1)-1, where P(n,k) = 2*k-1 if 2*k-1 <= F(n), P(n,k) = 2*(F(n)+1-k) if 2*k-1 > F(n).
EXAMPLE
Triangle begins:
k = 1 2 3 4 5 6 7 8
n=1: 1;
n=2: 2;
n=3: 3, 4;
n=4: 5, 7, 6;
n=5: 8, 10, 12, 11, 9;
n=6: 13, 15, 17, 19, 20, 18, 16, 14;
Subtracting F(n)-1 from each term in row n produces a permutation of 1 .. F(n):
1;
1;
1,2;
1,3,2;
1,3,5,4,2;
1,3,5,7,8,6,4,2;
...
MATHEMATICA
T[n_, k_]:=Module[{P, Result}, P= If[2*k-1 <=Fibonacci[n], 2*k-1, 2*(Fibonacci[n]+1-k)]; Result=P+Fibonacci[n+1]-1; Result]; Nmax=6; Table[T[n, k], {n, 1, Nmax}, {k, 1, Fibonacci[n]}]//Flatten
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Boris Putievskiy, Oct 17 2024
STATUS
approved