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A370655
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.
6
2, 1, 3, 4, 5, 7, 6, 8, 9, 10, 13, 14, 11, 12, 16, 15, 17, 20, 19, 18, 21, 26, 27, 24, 25, 22, 23, 29, 28, 30, 35, 32, 33, 34, 31, 36, 43, 44, 41, 42, 39, 40, 37, 38, 46, 45, 47, 54, 49, 52, 51, 50, 53, 48, 55
OFFSET
1,1
COMMENTS
Generalization of the Cantor numbering method for two adjacent diagonals. A pair of neighboring diagonals are combined into one block.
The sequence is a self-inverse permutation of natural numbers.
The sequence is an intra-block permutation of integer positive numbers.
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
FORMULA
Linear sequence:
a(n) = P(n) + (L(n)-1)*(2*L(n)-1), where L(n) = ceiling((sqrt(8*n+1)-1)/4),
L(n) = A204164(n),
R(n) = n - (L(n)-1)*(2*L(n)-1),
P(n) = -R(n) + 2*L(n)-2, if R(n) < 2*L(n) - 1 and R(n) mod 2 = 1, P(n) = -R(n) + 2*L(n), if R(n) < 2*L(n) - 1 and R(n) mod 2 = 0, P(n) = 2*L(n), if R(n) = 2*L(n) - 1, P(n) = R(n)-1, if R(n) = 2*L(n), P(n) = R(n), if R(n) > 2*L(n) and R(n) mod 2 = 1, P(n) = 6*L(n) - R(n), if R(n) > 2*L(n) and R(n) mod 2 = 0.
Triangular array T(n,k) for 1 <= k <= 4*n-1 (see Example):
T(n,k) = (n-1)*(2*n-1) + P(n,k), where
P(n,k) = 2*n-k-2 if k < 2*n-1 and k mod 2 = 1,
2*n-k if k < 2*n-1 and k mod 2 = 0,
2*k if k = 2*n-1,
k-1 if k = 2*n,
k if k > 2*n and k mod 2 = 1,
6*n-k if k > 2*n and k mod 2 = 0.
EXAMPLE
Triangle begins:
k = 1 2 3 4 5 6 7 8 9 10 11
n=1: 2, 1, 3;
n=2: 4, 5, 7, 6, 8, 9, 10;
n=3: 13, 14, 11, 12, 16, 15, 17, 20, 19, 18, 21;
Subtracting (n-1)*(2*n-1) from each term is row n is a self-inverse permutation of 1 .. 4*n-1,
2,1,3,
1,2,4,3,5,6,7,
3,4,1,2,6,5,7,10,9,8,11,
...
The triangle rows can be arranged as two successive upward antidiagonals in an array:
2, 3, 7, 10, 16, 21, ...
1, 5, 9, 12, 18, 23, ...
4, 8, 11, 19, 22, 34, ...
6, 14, 20, 25, 33, 40, ...
13, 17, 24, 32, 39, 51, ...
15, 27, 35, 42, 52, 61, ...
MATHEMATICA
Nmax = 21;
a[n_] := Module[{L, R, P, Result}, L = Ceiling[(Sqrt[8*n + 1] - 1)/4];
R = n - (L - 1)*(2*L - 1);
P = If[R < 2*L - 1, If[Mod[R, 2] == 1, -R + 2*L - 2, -R + 2*L],
If[R == 2*L - 1, 2*L,
If[R == 2*L, R - 1, If[Mod[R, 2] == 1, R, 6*L - R]]]];
Result = P + (L - 1)*(2*L - 1);
Result]
Table[a[n], {n, 1, Nmax}]
CROSSREFS
Sequence in context: A346786 A239469 A282840 * A082750 A048212 A282864
KEYWORD
nonn,tabf
AUTHOR
Boris Putievskiy, Feb 24 2024
STATUS
approved