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).
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).
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
KEYWORD
nonn,easy
AUTHOR
Donovan Young, Mar 09 2023
STATUS
approved