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A361378
Number of musical scales in n tone equal temperament respecting the property that alternate notes are 3 or 4 semitones apart.
1
0, 1, 2, 3, 3, 3, 8, 8, 12, 16, 25, 33, 45, 66, 91, 128, 177, 252, 351, 491, 689, 966, 1354, 1894, 2658, 3723, 5217, 7309, 10244, 14355, 20112, 28185, 39494, 55343, 77547, 108667, 152272, 213372, 298992, 418968, 587089, 822665, 1152777, 1615350
OFFSET
1,3
COMMENTS
If you take any three consecutive notes in the scales counted by a(n) (with cyclic identification) then the distance between the first and third is either 3 or 4 semitones. a(n) is also the number of subsets of Z/nZ that 1) contain 0; 2) contain no subset of the form {x,x+1,x+2}; 3) have no superset satisfying property 2).
FORMULA
a(n) = a(n-2) + a(n-3) + a(n-4) - a(n-6).
G.f.: x^2*(1+2*x+2*x^2-3*x^4)/(1-x^2-x^3-x^4+x^6).
EXAMPLE
For n=4 there are four notes, call them 0, 1, 2, and 3. The scales are 01, 02, and 03 and so a(4)=3.
MATHEMATICA
LinearRecurrence[{0, 1, 1, 1, 0, -1}, {0, 1, 2, 3, 3, 3}, 100]
CROSSREFS
Sequence in context: A088041 A334579 A199457 * A347208 A210483 A022296
KEYWORD
nonn,easy
AUTHOR
Donovan Young, Mar 09 2023
STATUS
approved