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A274879 A statistic on orbital systems over n sectors: the number of orbitals with k returns. 10
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, 80, 28, 56, 72, 64, 32, 252, 504, 672, 672, 480, 192, 84, 168, 224, 224, 160, 64, 924, 1848, 2520, 2688, 2240, 1344, 448, 264, 528, 720, 768, 640, 384, 128 (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.

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.

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.

A108747 is a subtriangle.

LINKS

Table of n, a(n) for n=0..56.

Peter Luschny, Orbitals

FORMULA

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).

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] [2, 4] 6

[ 4] [2, 4] 6

[ 5] [6, 12, 12] 30

[ 6] [4, 8, 8] 20

[ 7] [20, 40, 48, 32] 140

[ 8] [10, 20, 24, 16] 70

[ 9] [70, 140, 180, 160, 80] 630

[10] [28, 56, 72, 64, 32] 252

[11] [252, 504, 672, 672, 480, 192] 2772

T(6,0) = 4 because the following 4 orbitals stay above or below the central

circle: [-1, -1, -1, 1, 1, 1], [-1, -1, 1, -1, 1, 1], [1, 1, -1, 1, -1, -1],

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

PROG

(Sage)

# Brute force counting, function unit_orbitals defined in A274709.

def orbital_returns(n):

    if n == 0: return [1]

    S = [0]*((n+1)//2)

    for u in unit_orbitals(n):

        L = list(accumulate(u))

        Z = len(filter(lambda z: z == 0, L))

        S[Z-1] += 1  # exclude origin

    return S

for n in (0..10): print orbital_returns(n)

CROSSREFS

Cf. A056040 (row sum), A108747, A232500, A241543 (col. 0).

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).

Sequence in context: A035096 A066675 A219433 * A222043 A222153 A219403

Adjacent sequences:  A274876 A274877 A274878 * A274880 A274881 A274882

KEYWORD

nonn,tabf

AUTHOR

Peter Luschny, Jul 11 2016

STATUS

approved

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Last modified October 21 14:06 EDT 2019. Contains 328300 sequences. (Running on oeis4.)