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A208593
Number of n-bead necklaces labeled with numbers -4..4 not allowing reversal, with sum zero.
2
1, 5, 21, 125, 791, 5457, 39019, 288317, 2178929, 16773395, 131034839, 1036252649, 8279446917, 66733111919, 541954722471, 4430427981533, 36428763143945, 301074015186469, 2499725665085301, 20840038803521835, 174388665638906551, 1464205768804076875
OFFSET
1,2
LINKS
FORMULA
a(n) = (1/n) * Sum_{d | n} totient(n/d) * A025014(d). - Andrew Howroyd, Mar 02 2017
EXAMPLE
All solutions for n=3:
.-4...-2...-2...-3...-1...-3...-2...-3...-3...-4....0...-3...-2...-4...-1...-4
..2...-1....2....1....1....2....3...-1....3....1....0....0....0....0....0....3
..2....3....0....2....0....1...-1....4....0....3....0....3....2....4....1....1
..
.-1...-4...-3...-2...-2
.-1....4....4....1...-2
..2....0...-1....1....4
MATHEMATICA
comps[r_, m_, k_] := Sum[(-1)^i*Binomial[r - 1 - i*m, k - 1]*Binomial[k, i], {i, 0, Floor[(r - k)/m]}]; a[n_Integer, k_] := DivisorSum[n, EulerPhi[n/#] comps[#*(k + 1), 2 k + 1, #] &]/n; a[n_] = a[n, 4]; Array[a, 22] (* Jean-François Alcover, Nov 01 2017, after Andrew Howroyd *)
CROSSREFS
Column 4 of A208597.
Sequence in context: A218962 A124311 A353736 * A213009 A316102 A352388
KEYWORD
nonn
AUTHOR
R. H. Hardin, Feb 29 2012
EXTENSIONS
a(16)-a(22) from Andrew Howroyd, Mar 02 2017
STATUS
approved