A361378 Number of musical scales in n tone equal temperament respecting the property that alternate notes are 3 or 4 semitones apart.

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

Author

Donovan Young, Mar 09 2023

Keywords

• nonn
• easy

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).

Examples

			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.
		

Links

• Martin Epstein, mastodon.xyz thread, Feb 23 2023.
• Donovan Young, The number of scales with the natural thirds property.
• Index entries for linear recurrences with constant coefficients, signature (0,1,1,1,0,-1).

Programs

Mathematica

LinearRecurrence[{0,1,1,1,0,-1},{0,1,2,3,3,3},100]

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).