A274880 - OEIS

A274880

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

1, 1, 2, 5, 1, 4, 2, 18, 11, 1, 10, 8, 2, 65, 57, 17, 1, 28, 28, 12, 2, 238, 252, 116, 23, 1, 84, 96, 54, 16, 2, 882, 1050, 615, 195, 29, 1, 264, 330, 220, 88, 20, 2, 3300, 4257, 2915, 1210, 294, 35, 1, 858, 1144, 858, 416, 130, 24, 2, 12441, 17017, 13013, 6461, 2093, 413, 41, 1

(list; graph; refs; listen; history; text; internal format)

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.

A 'restart' of an orbital is a raise which starts from the central circle.

A118920 is a subtriangle.

LINKS

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

Peter Luschny, Orbitals

FORMULA

For even n>0: T(n,k) = 4*(k+1)*binomial(n,n/2-k-1)/n for k=0..n/2-1 (from A118920).

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] [5, 1] 6

[ 4] [4, 2] 6

[ 5] [18, 11, 1] 30

[ 6] [10, 8, 2] 20

[ 7] [65, 57, 17, 1] 140

[ 8] [28, 28, 12, 2] 70

[ 9] [238, 252, 116, 23, 1] 630

[10] [84, 96, 54, 16, 2] 252

[11] [882, 1050, 615, 195, 29, 1] 2772

T(6, 2) = 2 because there are two orbitals over 6 segments which have 2 ascents:

[-1, 1, 1, -1, 1, -1] and [1, -1, 1, -1, 1, -1].

PROG

(Sage) # uses[unit_orbitals from A274709]

from itertools import accumulate

# Brute force counting

def orbital_restart(n):

if n == 0: return [1]

S = [0]*((n+1)//2)

for u in unit_orbitals(n):

A = list(accumulate(u))

L = [1 if A[i] == 0 and A[i+1] == 1 else 0 for i in (0..n-2)]

S[sum(L)] += 1

return S

for n in (0..12): print(orbital_restart(n))

CROSSREFS

Cf. A056040 (row sum), A118920, A232500.

Other orbital statistics: A241477 (first zero crossing), A274706 (absolute integral), A274708 (peaks), A274709 (max. height), A274710 (number of turns), A274878 (span), A274879 (returns), A274881 (ascent).

Sequence in context: A316131 A153726 A229339 * A034005 A340407 A161688

Adjacent sequences: A274877 A274878 A274879 * A274881 A274882 A274883

KEYWORD

nonn,tabf

AUTHOR

Peter Luschny, Jul 11 2016

STATUS

approved