A377136 Triangle 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 row n is the (n+1)-st Fibonacci number for n > 0; see Comments.

1, 2, 3, 4, 5, 7, 6, 8, 10, 12, 11, 9, 13, 15, 17, 19, 20, 18, 16, 14, 21, 23, 25, 27, 29, 31, 33, 32, 30, 28, 26, 24, 22, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 53, 51, 49, 47, 45, 43, 41, 39, 37, 35, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 88, 86, 84, 82, 80, 78, 76, 74, 72, 70, 68, 66, 64, 62, 60, 58, 56, 89

Offset: 1

Author

Boris Putievskiy, Oct 17 2024

Keywords

• nonn
• tabf

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.

Examples

			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;
  ...
		

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.
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A000045, A194959, A375602, A376180.

Programs

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

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).