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A274710 A statistic on orbital systems over n sectors: the number of orbitals which make k turns. 10
1, 1, 0, 2, 0, 0, 6, 0, 2, 2, 2, 0, 0, 6, 12, 12, 0, 2, 4, 8, 4, 2, 0, 0, 6, 24, 52, 40, 18, 0, 2, 6, 18, 18, 18, 6, 2, 0, 0, 6, 36, 120, 180, 180, 84, 24, 0, 2, 8, 32, 48, 72, 48, 32, 8, 2, 0, 0, 6, 48, 216, 480, 744, 672, 432, 144, 30, 0, 2, 10, 50, 100, 200, 200, 200, 100, 50, 10, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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 'turn' of an orbital w takes place where signum(w[i]) is not equal to signum(w[i+1]).

A152659 is a subtriangle.

LINKS

Table of n, a(n) for n=0..78.

Peter Luschny, Orbitals

FORMULA

For even n>0: T(n,k) = 2*C(n/2-1,(k-1+mod(k-1,2))/2)*C(n/2-1,(k-1-mod(k-1,2))/2) for k=0..n-1 (from A152659).

EXAMPLE

Triangle read by rows, n>=0. The length of row n is n for n>=1.

[n] [k=0,1,2,...] [row sum]

[0] [1] 1

[1] [1] 1

[2] [0, 2] 2

[3] [0, 0, 6] 6

[4] [0, 2, 2,  2] 6

[5] [0, 0, 6, 12,  12] 30

[6] [0, 2, 4,  8,   4,   2] 20

[7] [0, 0, 6, 24,  52,  40,  18] 140

[8] [0, 2, 6, 18,  18,  18,   6,  2] 70

[9] [0, 0, 6, 36, 120, 180, 180, 84, 24] 630

T(5,2) = 6 because the six orbitals [-1, -1, 0, 1, 1], [-1, -1, 1, 1, 0], [0, -1, -1, 1, 1], [0, 1, 1, -1, -1], [1, 1, -1, -1, 0], [1, 1, 0, -1, -1] make 2 turns.

PROG

(Sage)

# Brute force counting, function unit_orbitals defined in A274709.

def orbital_turns(n):

    if n == 0: return [1]

    S = [0]*(n)

    for u in unit_orbitals(n):

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

        S[sum(L)] += 1

    return S

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

CROSSREFS

Cf. A056040 (row sum), A152659, A232500.

Other orbital statistics: A241477 (first zero crossing), A274706 (absolute integral), A274708 (number of peaks), A274709 (max. height), A274878 (span), A274879 (returns), A274880 (restarts), A274881 (ascent).

Sequence in context: A193474 A241020 A277444 * A028625 A221728 A278720

Adjacent sequences:  A274707 A274708 A274709 * A274711 A274712 A274713

KEYWORD

nonn,tabf

AUTHOR

Peter Luschny, Jul 10 2016

STATUS

approved

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Last modified November 15 11:18 EST 2019. Contains 329144 sequences. (Running on oeis4.)