A274708 A statistic on orbital systems over n sectors: the number of orbitals with k peaks.

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, 70, 104, 60, 16, 2, 384, 1026, 912, 372, 72, 6, 192, 350, 260, 100, 20, 2, 1214, 3727, 4095, 2220, 650, 99, 7, 534, 1152, 1050, 520, 150, 24, 2, 3822, 13200, 17178, 11600, 4510, 1032, 130, 8

Offset

0,3

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.

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]).

A097692 is a subtriangle.

Links

Table of n, a(n) for n=0..64.

Peter Luschny, Orbitals

Example

Triangle read by rows, n>=0. The length of row n is floor((n+1)/2) for n>=1.
[ n] [k=0,1,2,...]               [row sum]
[ 0] [  1]                           1
[ 1] [  1]                           1
[ 2] [  2]                           2
[ 3] [  4,    2]                     6
[ 4] [  4,    2]                     6
[ 5] [ 12,   15,   3]               30
[ 6] [ 10,    8,   2]               20
[ 7] [ 38,   68,  30,   4]         140
[ 8] [ 26,   30,  12,   2]          70
[ 9] [121,  272, 183,  49,  5]     630
[10] [ 70,  104,  60,  16,  2]     252
[11] [384, 1026, 912, 372, 72, 6] 2772
[12] [192,  350, 260, 100, 20, 2]  924
T(6, 2) = 2 because the two orbitals [-1, 1, -1, 1, -1, 1] and [1, -1, 1, -1, 1, -1] have 2 peaks.

Prog

(Sage) # uses[unit_orbitals from A274709]
# Brute force counting
def orbital_peaks(n):
    if n == 0: return [1]
    S = [0]*((n+1)//2)
    for u in unit_orbitals(n):
        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)]
        S[sum(L)] += 1
    return S
for n in (0..12): print(orbital_peaks(n))

Crossrefs

Cf. A025565 (even col. 0), A056040 (row sum), A097692, A232500.

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).

Sequence in context: A086152 A194577 A334970 * A280638 A333725 A163894

Adjacent sequences: A274705 A274706 A274707 * A274709 A274710 A274711

Keyword

nonn,tabf

Author

Peter Luschny, Jul 10 2016

Status

approved