A190719 Numbers that are congruent to {0, 1, 3, 5, 7, 8, 11} mod 12.

0, 1, 3, 5, 7, 8, 11, 12, 13, 15, 17, 19, 20, 23, 24, 25, 27, 29, 31, 32, 35, 36, 37, 39, 41, 43, 44, 47, 48, 49, 51, 53, 55, 56, 59, 60, 61, 63, 65, 67, 68, 71, 72, 73, 75, 77, 79, 80, 83, 84, 85, 87, 89, 91, 92, 95, 96, 97, 99, 101, 103, 104, 107, 108, 109, 111

Offset: 1

Author

Roberto Bertocco, May 29 2011

Keywords

• nonn
• easy

Comments

The key-numbers of the pitches of a minor neapolitan scale on a standard chromatic keyboard, with root = 0.

This sequence contains all odd primes. - Jonathan Vos Post, Jun 09 2011

Links

• Wikipedia, Neapolitan scale
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1,-1).

Crossrefs

Cf. A190785.

Programs

Magma

[n : n in [0..150] | n mod 12 in [0, 1, 3, 5, 7, 8, 11]]; // Wesley Ivan Hurt, Jul 21 2016

Maple

A190719:=n->12*floor(n/7)+[0, 1, 3, 5, 7, 8, 11][(n mod 7)+1]: seq(A190719(n), n=0..100); # Wesley Ivan Hurt, Jul 21 2016

Mathematica

Select[Range[0,120], MemberQ[{0,1,3,5,7,8,11}, Mod[#,12]]&] (* Harvey P. Dale, Jun 10 2011 *)

Formula

a(n) = a(n-1) + a(n-7) - a(n-8) for n>8.

G.f.: x^2*(1+2*x+2*x^2+2*x^3+x^4+3*x^5+x^6) / ( (x^6+x^5+x^4+x^3+x^2+x+1)*(x-1)^2 ). - R. J. Mathar, Jun 11 2011

a(n) = floor(12*n/7) - floor((n mod 7)/6) - floor(((n+3) mod 7)/5). - Rolf Pleisch, Jun 12 2011

From Wesley Ivan Hurt, Jul 21 2016: (Start)

a(n) = a(n-7) + 12 for n>7.

a(n) = (84*n - 91 - 9*(n mod 7) + 5*((n+1) mod 7) - 2*((n+2) mod 7) - 2*((n+3) mod 7) - 2*((n+4) mod 7) + 5*((n+5) mod 7) + 5*((n+6) mod 7))/49.

a(7k) = 12k-1, a(7k-1) = 12k-4, a(7k-2) = 12k-5, a(7k-3) = 12k-7, a(7k-4) = 12k-9, a(7k-5) = 12k-11, a(7k-6) = 12k-12. (End)