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A274878
A statistic on orbital systems over n sectors: the number of orbitals with span k.
10
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.
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
KEYWORD
nonn,tabf
AUTHOR
Peter Luschny, Jul 10 2016
STATUS
approved