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%I #27 Mar 27 2020 06:58:01
%S 1,1,2,4,2,4,2,12,15,3,10,8,2,38,68,30,4,26,30,12,2,121,272,183,49,5,
%T 70,104,60,16,2,384,1026,912,372,72,6,192,350,260,100,20,2,1214,3727,
%U 4095,2220,650,99,7,534,1152,1050,520,150,24,2,3822,13200,17178,11600,4510,1032,130,8
%N A statistic on orbital systems over n sectors: the number of orbitals with k peaks.
%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 An orbital w has a 'peak' at i+1 when signum(w[i]) < signum(w[i+1]) and signum(w[i+1]) > signum(w[i+2]).
%C A097692 is a subtriangle.
%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+1)/2) for n>=1.
%e [ n] [k=0,1,2,...] [row sum]
%e [ 0] [ 1] 1
%e [ 1] [ 1] 1
%e [ 2] [ 2] 2
%e [ 3] [ 4, 2] 6
%e [ 4] [ 4, 2] 6
%e [ 5] [ 12, 15, 3] 30
%e [ 6] [ 10, 8, 2] 20
%e [ 7] [ 38, 68, 30, 4] 140
%e [ 8] [ 26, 30, 12, 2] 70
%e [ 9] [121, 272, 183, 49, 5] 630
%e [10] [ 70, 104, 60, 16, 2] 252
%e [11] [384, 1026, 912, 372, 72, 6] 2772
%e [12] [192, 350, 260, 100, 20, 2] 924
%e T(6, 2) = 2 because the two orbitals [-1, 1, -1, 1, -1, 1] and [1, -1, 1, -1, 1, -1] have 2 peaks.
%o (Sage) # uses[unit_orbitals from A274709]
%o # Brute force counting
%o def orbital_peaks(n):
%o if n == 0: return [1]
%o S = [0]*((n+1)//2)
%o for u in unit_orbitals(n):
%o L = [1 if sgn(u[i]) < sgn(u[i+1]) and sgn(u[i+1]) > sgn(u[i+2]) else 0 for i in (0..n-3)]
%o S[sum(L)] += 1
%o return S
%o for n in (0..12): print(orbital_peaks(n))
%Y Cf. A025565 (even col. 0), A056040 (row sum), A097692, A232500.
%Y Other orbital statistics: A241477 (first zero crossing), A274706 (absolute integral), A274709 (max. height), A274710 (number of turns), A274878 (span), A274879 (returns), A274880 (restarts), A274881 (ascent).
%K nonn,tabf
%O 0,3
%A _Peter Luschny_, Jul 10 2016
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