cp's OEIS Frontend

A274879 A statistic on orbital systems over n sectors: the number of orbitals with k returns.

%I A274879 #28 Mar 23 2020 13:28:13
%S A274879 1,1,2,2,4,2,4,6,12,12,4,8,8,20,40,48,32,10,20,24,16,70,140,180,160,
%T A274879 80,28,56,72,64,32,252,504,672,672,480,192,84,168,224,224,160,64,924,
%U A274879 1848,2520,2688,2240,1344,448,264,528,720,768,640,384,128
%N A274879 A statistic on orbital systems over n sectors: the number of orbitals with k returns.
%C A274879 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 A274879 When a segment of an orbital starts at a point on the central circle this point is called a 'return' of the orbital if it is not the origin.
%C A274879 If an orbital touches the central circle only in the origin it is called a prime orbital. Column 0 counts the prime orbitals over n sectors.
%C A274879 A108747 is a subtriangle.
%H A274879 Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/Orbitals">Orbitals</a>
%F A274879 For even n>0: T(n,k) = 2^(k+1)*(k+1)*binomial(n-k-1,n/2)/(n-k-1) for k=0..n/2-1 (from A108747).
%e A274879 Triangle read by rows, n>=0. The length of row n is floor((n+1)/2) for n>=1.
%e A274879 [ n] [k=0,1,2,...] [row sum]
%e A274879 [ 0] [1] 1
%e A274879 [ 1] [1] 1
%e A274879 [ 2] [2] 2
%e A274879 [ 3] [2, 4] 6
%e A274879 [ 4] [2, 4] 6
%e A274879 [ 5] [6, 12, 12] 30
%e A274879 [ 6] [4, 8, 8] 20
%e A274879 [ 7] [20, 40, 48, 32] 140
%e A274879 [ 8] [10, 20, 24, 16] 70
%e A274879 [ 9] [70, 140, 180, 160, 80] 630
%e A274879 [10] [28, 56, 72, 64, 32] 252
%e A274879 [11] [252, 504, 672, 672, 480, 192] 2772
%e A274879 T(6,0) = 4 because the following 4 orbitals stay above or below the central
%e A274879 circle: [-1, -1, -1, 1, 1, 1], [-1, -1, 1, -1, 1, 1], [1, 1, -1, 1, -1, -1],
%e A274879 [1, 1, 1, -1, -1, -1].
%o A274879 (Sage) # uses[unit_orbitals from A274709]
%o A274879 from itertools import accumulate
%o A274879 # Brute force counting
%o A274879 def orbital_returns(n):
%o A274879     if n == 0: return [1]
%o A274879     S = [0]*((n+1)//2)
%o A274879     for u in unit_orbitals(n):
%o A274879         L = list(accumulate(u))
%o A274879         Z = len(list(filter(lambda z: z == 0, L)))
%o A274879         S[Z-1] += 1  # exclude origin
%o A274879     return S
%o A274879 for n in (0..10): print(orbital_returns(n))
%Y A274879 Cf. A056040 (row sum), A108747, A232500, A241543 (col. 0).
%Y A274879 Other orbital statistics: A241477 (first zero crossing), A274706 (absolute integral), A274708 (peaks), A274709 (max. height), A274710 (number of turns), A274878 (span), A274880 (restarts), A274881 (ascent).
%K A274879 nonn,tabf
%O A274879 0,3
%A A274879 _Peter Luschny_, Jul 11 2016