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%I #26 Mar 23 2020 17:33:36
%S 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,
%T 288,72,0,2,60,120,60,10,0,22,660,1320,660,110,0,2,124,390,300,96,12,
%U 0,26,1612,5070,3900,1248,156,0,2,252,1176,1260,588,140,14
%N A statistic on orbital systems over n sectors: the number of orbitals with span k.
%C 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.
%C The 'span' of an orbital w is the difference between the highest and the lowest level of the orbital system touched by w.
%H Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/Orbitals">Orbitals</a>
%e Triangle read by rows, n>=0. The length of row n is floor((n+2)/2).
%e [ n] [k=0,1,2,...] [row sum]
%e [ 0] [1] 1
%e [ 1] [1] 1
%e [ 2] [0, 2] 2
%e [ 3] [0, 6] 6
%e [ 4] [0, 2, 4] 6
%e [ 5] [0, 10, 20] 30
%e [ 6] [0, 2, 12, 6] 20
%e [ 7] [0, 14, 84, 42] 140
%e [ 8] [0, 2, 28, 32, 8] 70
%e [ 9] [0, 18, 252, 288, 72] 630
%e [10] [0, 2, 60, 120, 60, 10] 252
%e T(6, 3) = 6 because the span of the following six orbitals is 3:
%e [-1, -1, -1, 1, 1, 1], [-1, -1, 1, 1, 1, -1], [-1, 1, 1, 1, -1, -1],
%e [1, -1, -1, -1, 1, 1], [1, 1, -1, -1, -1, 1], [1, 1, 1, -1, -1, -1].
%o (Sage) # uses[unit_orbitals from A274709]
%o from itertools import accumulate
%o # Brute force counting.
%o def orbital_span(n):
%o if n == 0: return [1]
%o S = [0]*((n+2)//2)
%o for u in unit_orbitals(n):
%o L = list(accumulate(u))
%o S[max(L) - min(L)] += 1
%o return S
%o for n in (0..10): print(orbital_span(n))
%Y Cf. A056040 (row sum), A232500.
%Y 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).
%K nonn,tabf
%O 0,4
%A _Peter Luschny_, Jul 10 2016