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A374447 Inverse permutation to A373498.

%I A374447 #17 Aug 04 2024 13:35:05
%S A374447 2,1,3,9,4,7,6,8,5,10,20,11,18,13,16,15,17,12,19,14,21,35,22,33,24,31,
%T A374447 26,29,28,30,23,32,25,34,27,36,54,37,52,39,50,41,48,43,46,45,47,38,49,
%U A374447 40,51,42,53,44,55
%N A374447 Inverse permutation to A373498.
%C A374447 Triangle read by rows where row n is a block of length 4*n-1 which is a permutation of the numbers of its constituents.
%C A374447 Generalization of the Cantor numbering method for two adjacent diagonals. A pair of neighboring diagonals are combined into one block.
%C A374447 The sequence is an intra-block permutation of positive integers.
%C A374447 The sequence A373498 generates the cyclic group C6 under composition. The elements of C6 are the successive compositions of A373498 with itself:  A374494 = A373498(A373498) = A373498^2, A370655 = A373498^3, A374531 = A373498^4, A374447 = A373498^5. The identity element is A000027 = A373498^6. - _Boris Putievskiy_, Aug 03 2024
%H A374447 Boris Putievskiy, <a href="/A374447/b374447.txt">Table of n, a(n) for n = 1..9870</a>
%H A374447 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 A374447 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A374447 Linear sequence:
%F A374447 a(n) = P(n) + (L(n)-1)*(2*L(n)-1), where L(n) = ceiling((sqrt(8*n+1)-1)/4),
%F A374447 L(n) = A204164(n), P(n) = 4*L(n) - R(n) - 1, if R(n) <= 2*L(n) and R(n) mod 2 = 1, P(n) = R(n) - 1, if R(n) <= 2*L(n) and R(n) mod 2 = 0, P(n) = R(n), if R(n) > 2*L(n) and R(n) mod 2 = 1, P(n) = - 2*L(n) + R(n), if R(n) > 2*L(n) and R(n) mod 2 = 0.
%F A374447 a(n) = A373498(A373498(A373498(A373498(A373498(n))))).
%F A374447 Triangular array T(n,k) for 1 <= k <= 4*n-1 (see Example):
%F A374447 T(n,k) = (n-1)*(2*n-1) + P(n,k), where P(n,k) = 4*n - k - 1, if k <= 2*n and k mod 2 = 1, P(n,k) = k-1, if k <= 2*n and k mod 2 = 0, P(n,k) = k, if k >  2*n and k mod 2 = 1, P(n,k) = -2*n + k, if k >  2*n and k mod 2 = 0.
%e A374447 Triangle begins:
%e A374447      k = 1   2   3   4   5   6   7   8   9  10  11
%e A374447   n=1:   2,  1,  3;
%e A374447   n=2:   9,  4,  7,  6,  8,  5, 10;
%e A374447   n=3:  20, 11, 18, 13, 16, 15, 17, 12, 19, 14, 21;
%e A374447 The triangle's rows can be arranged as two successive upward antidiagonals in an array:
%e A374447     2,  3,  7, 10, 16, 21, ...
%e A374447     1,  4,  5, 13, 14, 26, ...
%e A374447     9,  8, 18, 19, 31, 34, ...
%e A374447     6, 11, 12, 24, 25, 41, ...
%e A374447    20, 17, 33, 32, 50, 51, ...
%e A374447    15, 22, 23, 39, 40. 60, ...
%e A374447 Subtracting (n-1)*(2*n-1) from each term in row n is a permutation of 1 .. 4*n-1:
%e A374447     2,1,3,
%e A374447     6,1,4,3,5,2,7,
%e A374447    10,1,8,3,6,5,7,2,9,4,11
%e A374447    ...
%e A374447 The inverse permutation of each permutation in example A373498 is equal to the corresponding permutation above:
%e A374447    (2,1,3)^(-1) = (2,1,3),
%e A374447    (2,6,4,3,5,1,7)^(-1) = (6,1,4,3,5,2,7),
%e A374447    (2,8,4,10,6,5,7,3,9,1,11)^(-1) = (10,1,8,3,6,5,7,2,9,4,11).
%e A374447 The 5th power of each permutation in example A373498 is equal to the corresponding permutation above:
%e A374447    (2,1,3)^5 = (2,1,3),
%e A374447    (2,6,4,3,5,1,7)^5 = (6,1,4,3,5,2,7),
%e A374447    (2,8,4,10,6,5,7,3,9,1,11)^5 = (10,1,8,3,6,5,7,2,9,4,11).
%t A374447 Nmax=21;
%t A374447 a[n_]:=Module[{L,R,P,Result},L=Ceiling[(Sqrt[8*n+1]-1)/4];
%t A374447 R=n-(L-1)*(2*L-1); P=Which[R<=2*L&&Mod[R,2]==1,4*L-R-1,R<=2*L&&Mod[R,2]==0,R-1,R>2*L&&Mod[R,2]==1,R,R>2*L&&Mod[R,2]==0,-2*L+R];
%t A374447 Result=P+(L-1)*(2*L-1);
%t A374447 Result]
%t A374447 Table[a[n],{n,1,Nmax}]
%t A374447 Nmax=21;
%t A374447 a[n_]:=Module[{L,R,P,Result},L=Ceiling[(Sqrt[8*n+1]-1)/4];
%t A374447 R=n-(L-1)*(2*L-1); P=Which[R<=2*L-1&&Mod[R,2]==1,R+1,R<=2*L-1&&Mod[R,2]==0,R+2*L,R>2*L-1&&Mod[R,2]==1,R,R>2*L-1&&Mod[R,2]==0,4*L-1-R];
%t A374447 Result=P+(L-1)*(2*L-1);
%t A374447 Result]
%t A374447 Table[a[n],{n,1,Nmax}] (* A373498 *)
%t A374447 Table[a[a[a[a[a[n]]]]],{n,1,Nmax}] (* this sequence *)
%Y A374447 Cf. A000027, A004767 (row lengths), A204164, A370655, A373498, A374494, A374531.
%K A374447 nonn,tabf
%O A374447 1,1
%A A374447 _Boris Putievskiy_, Jul 08 2024