cp's OEIS Frontend

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

%I A274878 #26 Mar 23 2020 17:33:36
%S A274878 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 A274878 288,72,0,2,60,120,60,10,0,22,660,1320,660,110,0,2,124,390,300,96,12,
%U A274878 0,26,1612,5070,3900,1248,156,0,2,252,1176,1260,588,140,14
%N A274878 A statistic on orbital systems over n sectors: the number of orbitals with span k.
%C A274878 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 A274878 The 'span' of an orbital w is the difference between the highest and the lowest level of the orbital system touched by w.
%H A274878 Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/Orbitals">Orbitals</a>
%e A274878 Triangle read by rows, n>=0. The length of row n is floor((n+2)/2).
%e A274878 [ n] [k=0,1,2,...]          [row sum]
%e A274878 [ 0] [1]                        1
%e A274878 [ 1] [1]                        1
%e A274878 [ 2] [0, 2]                     2
%e A274878 [ 3] [0, 6]                     6
%e A274878 [ 4] [0, 2, 4]                  6
%e A274878 [ 5] [0, 10, 20]               30
%e A274878 [ 6] [0, 2, 12, 6]             20
%e A274878 [ 7] [0, 14, 84, 42]          140
%e A274878 [ 8] [0, 2, 28, 32, 8]         70
%e A274878 [ 9] [0, 18, 252, 288, 72]    630
%e A274878 [10] [0, 2, 60, 120, 60, 10]  252
%e A274878 T(6, 3) = 6 because the span of the following six orbitals is 3:
%e A274878 [-1, -1, -1, 1, 1, 1], [-1, -1, 1, 1, 1, -1], [-1, 1, 1, 1, -1, -1],
%e A274878 [1, -1, -1, -1, 1, 1], [1, 1, -1, -1, -1, 1], [1, 1, 1, -1, -1, -1].
%o A274878 (Sage) # uses[unit_orbitals from A274709]
%o A274878 from itertools import accumulate
%o A274878 # Brute force counting.
%o A274878 def orbital_span(n):
%o A274878     if n == 0: return [1]
%o A274878     S = [0]*((n+2)//2)
%o A274878     for u in unit_orbitals(n):
%o A274878         L = list(accumulate(u))
%o A274878         S[max(L) - min(L)] += 1
%o A274878     return S
%o A274878 for n in (0..10): print(orbital_span(n))
%Y A274878 Cf. A056040 (row sum), A232500.
%Y A274878 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 A274878 nonn,tabf
%O A274878 0,4
%A A274878 _Peter Luschny_, Jul 10 2016