A274878 A statistic on orbital systems over n sectors: the number of orbitals with span k.

1, 1, 0, 2, 0, 6, 0, 2, 4, 0, 10, 20, 0, 2, 12, 6, 0, 14, 84, 42, 0, 2, 28, 32, 8, 0, 18, 252, 288, 72, 0, 2, 60, 120, 60, 10, 0, 22, 660, 1320, 660, 110, 0, 2, 124, 390, 300, 96, 12, 0, 26, 1612, 5070, 3900, 1248, 156, 0, 2, 252, 1176, 1260, 588, 140, 14

Offset

0,4

Comments

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.

The 'span' of an orbital w is the difference between the highest and the lowest level of the orbital system touched by w.

Links

Table of n, a(n) for n=0..63.

Peter Luschny, Orbitals

Example

Triangle read by rows, n>=0. The length of row n is floor((n+2)/2).
[ n] [k=0,1,2,...]          [row sum]
[ 0] [1]                        1
[ 1] [1]                        1
[ 2] [0, 2]                     2
[ 3] [0, 6]                     6
[ 4] [0, 2, 4]                  6
[ 5] [0, 10, 20]               30
[ 6] [0, 2, 12, 6]             20
[ 7] [0, 14, 84, 42]          140
[ 8] [0, 2, 28, 32, 8]         70
[ 9] [0, 18, 252, 288, 72]    630
[10] [0, 2, 60, 120, 60, 10]  252
T(6, 3) = 6 because the span of the following six orbitals is 3:
[-1, -1, -1, 1, 1, 1], [-1, -1, 1, 1, 1, -1], [-1, 1, 1, 1, -1, -1],
[1, -1, -1, -1, 1, 1], [1, 1, -1, -1, -1, 1], [1, 1, 1, -1, -1, -1].

Prog

(Sage) # uses[unit_orbitals from A274709]
from itertools import accumulate
# Brute force counting.
def orbital_span(n):
    if n == 0: return [1]
    S = [0]*((n+2)//2)
    for u in unit_orbitals(n):
        L = list(accumulate(u))
        S[max(L) - min(L)] += 1
    return S
for n in (0..10): print(orbital_span(n))

Crossrefs

Cf. A056040 (row sum), A232500.

Other orbital statistics: A241477 (first zero crossing), A274706 (absolute integral), A274708 (number of peaks), A274709 (max. height), A274710 (number of turns), A274879 (returns), A274880 (restarts), A274881 (ascent).

Sequence in context: A138701 A332400 A355143 * A050821 A076257 A274881

Adjacent sequences: A274875 A274876 A274877 * A274879 A274880 A274881

Keyword

nonn,tabf

Author

Peter Luschny, Jul 10 2016

Status

approved