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A376180
Triangle read by rows (blocks). Each row consists of a permutation of the numbers of its constituents. The length of row number n is the n-th pentagonal number n(3n-1)/2 = A000326(n); see Comments.
1
1, 4, 5, 3, 6, 2, 13, 12, 14, 11, 15, 10, 16, 9, 17, 8, 18, 7, 30, 29, 31, 28, 32, 27, 33, 26, 34, 25, 35, 24, 36, 23, 37, 22, 38, 21, 39, 20, 40, 19, 58, 59, 57, 60, 56, 61, 55, 62, 54, 63, 53, 64, 52, 65, 51, 66, 50, 67, 49, 68, 48, 69, 47, 70, 46, 71, 45, 72, 44, 73, 43, 74, 42, 75, 41, 101, 102, 100, 103, 99, 104, 98, 105, 97, 106, 96, 107
OFFSET
1,2
COMMENTS
A209278 presents an algorithm for generating permutations.
The sequence is an intra-block permutation of integer positive numbers.
FORMULA
Linear sequence:
a(n) = P(n) + (L(n)-1)^2*L(n)/2. a(n) = P(n) + A002411(L(n)-1), where P = (L(n)(3L(n) - 1)/2 - R(n) + 2)/2 if R(n) is odd and L(n)(3L(n) - 1)/2 is odd, P = (R(n) + L(n)(3L(n) - 1)/2 + 1)/2 if R(n) is odd and L(n)(3L(n) - 1)/2 is even, P = ceiling((L(n)(3L(n) - 1)/2 + 1)/2) + R(n)/2 if R(n) is even and L(n)(3L(n) - 1)/2 is odd, P = ceiling((L(n)(3L(n) - 1)/2 + 1)/2) - R(n)/2 if R(n) is even and L(n)(3L(n) - 1)/2 is even. L(n) = ceiling(x(n)), x(n) is largest real root of the equation x^2*(x+1)-2*n = 0.
Triangular array T(n,k) for 1 <= k <= n(3n-1)/2 (see Example):
T(n,k) = P(n,k) + (n-1)^2*n/2, T(n,k) = P(n,k) + A002411(n-1), where P(n,k) = (n(3n - 1)/2 - k + 2)/2 if k is odd and n(3n - 1)/2 is odd,
P(n,k) = (k + n(3n - 1)/2 + 1)/2 if k is odd and n(3n - 1)/2 is even, P(n,k) = ceiling((n(3n - 1)/2 + 1)/2) + k/2 if k is even and n(3n - 1)/2 is odd, P(n,k) = ceiling((n(3n - 1)/2 + 1)/2) - k/2 if k is even and n(3n - 1)/2 is even.
EXAMPLE
Triangle begins:
k = 1 2 3 4 5 6 7 8 9 10 11 12
n=1: 1;
n=2: 4, 5, 3, 6, 2;
n=3: 13, 12, 14, 11, 15, 10, 16, 9, 17, 8, 18, 7;
Subtracting (n-1)^2*n/2 from each term in row n is a permutation of 1 .. n(3n-1)/2:
1;
3,4,2,5,1;
7,6,8,5,9,4,10,3,11,2,12,1;
MATHEMATICA
a[n_]:=Module[{L, R, P, Result}, L=Ceiling[Max[x/.NSolve[x^2 (x+1)-2 n==0, x, Reals]]];
R=n-((L-1)^2)*L/2; P=Which[OddQ[R]&&OddQ[L*(3*L-1)/2], (L*(3*L-1)/2-R+2)/2, OddQ[R]&&EvenQ[L*(3*L-1)/2], (R+L*(3*L-1)/2+1)/2, EvenQ[R]&&OddQ[L*(3*L-1)/2], Ceiling[(L*(3*L-1)/2+1)/2]+R/2, EvenQ[R]&&EvenQ[L*(3*L-1)/2], Ceiling[(L*(3*L-1)/2+1)/2]-R/2 ];
Result=P+(L-1)^2*L/2; Result]
Nmax=18; Table[a[n], {n, 1, Nmax}]
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Boris Putievskiy, Sep 14 2024
STATUS
approved