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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.

%I A377136 #25 Nov 15 2024 01:18:58
%S A377136 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,
%T A377136 33,32,30,28,26,24,22,34,36,38,40,42,44,46,48,50,52,54,53,51,49,47,45,
%U A377136 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
%N 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.
%C A377136 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.
%C A377136 The sequence is an intra-block permutation of the positive integers.
%H A377136 Boris Putievskiy, <a href="/A377136/b377136.txt">Table of n, a(n) for n = 1..6764</a>
%H A377136 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 A377136 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A377136 T(n,k) for 1 <= k <= F(n) (see Example):
%F A377136 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).
%e A377136 Triangle begins:
%e A377136      k =  1   2   3   4   5   6   7   8
%e A377136   n=1:    1;
%e A377136   n=2:    2;
%e A377136   n=3:    3,  4;
%e A377136   n=4:    5,  7,  6;
%e A377136   n=5:    8, 10, 12, 11,  9;
%e A377136   n=6:   13, 15, 17, 19, 20, 18, 16, 14;
%e A377136 Subtracting F(n)-1 from each term in row n produces a permutation of 1 .. F(n):
%e A377136   1;
%e A377136   1;
%e A377136   1,2;
%e A377136   1,3,2;
%e A377136   1,3,5,4,2;
%e A377136   1,3,5,7,8,6,4,2;
%e A377136   ...
%t A377136 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
%Y A377136 Cf. A000045, A194959, A375602, A376180.
%K A377136 nonn,tabf
%O A377136 1,2
%A A377136 _Boris Putievskiy_, Oct 17 2024