A152659 -id:A152659 - cp's OEIS frontend

A274710 A statistic on orbital systems over n sectors: the number of orbitals which make k turns.

1, 1, 0, 2, 0, 0, 6, 0, 2, 2, 2, 0, 0, 6, 12, 12, 0, 2, 4, 8, 4, 2, 0, 0, 6, 24, 52, 40, 18, 0, 2, 6, 18, 18, 18, 6, 2, 0, 0, 6, 36, 120, 180, 180, 84, 24, 0, 2, 8, 32, 48, 72, 48, 32, 8, 2, 0, 0, 6, 48, 216, 480, 744, 672, 432, 144, 30, 0, 2, 10, 50, 100, 200, 200, 200, 100, 50, 10, 2

Offset: 0

Peter Luschny, Jul 10 2016

The definition of an orbital system is given in A232500 (see also the illustration there). The number of orbitals over n sectors is counted by the swinging factorial A056040.
A 'turn' of an orbital w takes place where signum(w[i]) is not equal to signum(w[i+1]).
A152659 is a subtriangle.

			Triangle read by rows, n>=0. The length of row n is n for n>=1.
[n] [k=0,1,2,...]                      [row sum]
[0] [1]                                    1
[1] [1]                                    1
[2] [0, 2]                                 2
[3] [0, 0, 6]                              6
[4] [0, 2, 2,  2]                          6
[5] [0, 0, 6, 12,  12]                    30
[6] [0, 2, 4,  8,   4,   2]               20
[7] [0, 0, 6, 24,  52,  40,  18]         140
[8] [0, 2, 6, 18,  18,  18,   6,  2]      70
[9] [0, 0, 6, 36, 120, 180, 180, 84, 24] 630
T(5,2) = 6 because the six orbitals [-1, -1, 0, 1, 1], [-1, -1, 1, 1, 0], [0, -1, -1, 1, 1], [0, 1, 1, -1, -1], [1, 1, -1, -1, 0], [1, 1, 0, -1, -1] make 2 turns.

Peter Luschny, Orbitals

Cf. A056040 (row sum), A152659, A232500.

Other orbital statistics: A241477 (first zero crossing), A274706 (absolute integral), A274708 (number of peaks), A274709 (max. height), A274878 (span), A274879 (returns), A274880 (restarts), A274881 (ascent).

# uses[unit_orbitals from A274709]
# Brute force counting
def orbital_turns(n):
    if n == 0: return [1]
    S = [0]*(n)
    for u in unit_orbitals(n):
        L = sum(0 if sgn(u[i]) == sgn(u[i+1]) else 1 for i in (0..n-2))
        S[L] += 1
    return S
for n in (0..12): print(orbital_turns(n))

For even n>0: T(n,k) = 2*C(n/2-1,(k-1+mod(k-1,2))/2)*C(n/2-1,(k-1-mod(k-1,2))/2) for k=0..n-1 (from A152659).

A375763 Irregular triangle read by rows, T(n,k) is the number of North-East lattice paths from (0,0) to (n,n+2) that stay weakly above y = x, with weight = k + A000217(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 3, 4, 5, 4, 4, 3, 2, 1, 1, 1, 4, 7, 10, 11, 11, 11, 9, 8, 6, 5, 3, 2, 1, 1, 1, 5, 11, 18, 24, 27, 30, 29, 28, 25, 23, 19, 16, 12, 10, 7, 5, 3, 2, 1, 1, 1, 6, 16, 30, 46, 59, 71, 78, 81, 81, 78, 74, 67, 60, 52, 46, 37, 31, 24

Offset: 0

John Tyler Rascoe, Aug 26 2024

Here the weight of a lattice path is the area under the path and above the x-axis. T(n,k) also counts the number of integer compositions of (3*n) + (2*k) + 6 with adjacent differences in {-1,1}, first part 1, and last part 3.

			Triangle begins:
    k=0  1  2   3   4   5   6   7   8   9  10  11  12  13  14
 n=0: 1;
 n=1: 1, 1, 1;
 n=2: 1, 2, 2,  2,  1,  1;
 n=3: 1, 3, 4,  5,  4,  4,  3,  2,  1,  1;
 n=4: 1, 4, 7, 10, 11, 11, 11,  9,  8,  6,  5,  3,  2,  1,  1;
 ...
T(1,0) = 1: (NENN).
T(2,1) = 2: (NNEENN) and (NENNEN).
T(3,2) = 4: (NENENNNE), (NENNENEN), (NNEENNEN), and (NNENEENN).

John Tyler Rascoe, Python program.

Cf. A000245 (empirical row sums), A000217 (row lengths).
Cf. A227543 (paths of this kind from (0,0) to (n,n), offset 1 for (0,0) to (n,n+1)).
Cf. A000108, A152659, A173258, A227543, A268429.

Python
```
# see linked program
```

cp's OEIS Frontend

A274710 A statistic on orbital systems over n sectors: the number of orbitals which make k turns.

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