A191275 Numbers that are congruent to {0, 1, 3, 5, 7, 9, 11} mod 12.

0, 1, 3, 5, 7, 9, 11, 12, 13, 15, 17, 19, 21, 23, 24, 25, 27, 29, 31, 33, 35, 36, 37, 39, 41, 43, 45, 47, 48, 49, 51, 53, 55, 57, 59, 60, 61, 63, 65, 67, 69, 71, 72, 73, 75, 77, 79, 81, 83, 84, 85, 87, 89, 91, 93, 95, 96, 97, 99, 101, 103, 105, 107, 108, 109, 111

Offset: 1

Author

Roberto Bertocco, May 29 2011

Keywords

• nonn
• easy

Comments

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

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, 9, 11]]; // Wesley Ivan Hurt, Jul 21 2016

Maple

A191275:=n->12*floor(n/7)+[0, 1, 3, 5, 7, 9, 11][(n mod 7)+1]: seq(A191275(n), n=0..100); # Wesley Ivan Hurt, Jul 21 2016

Mathematica

Select[Range[0,120], MemberQ[{0,1,3,5,7,9,11}, Mod[#,12]]&] (* or *) LinearRecurrence[{1,0,0,0,0,0,1,-1}, {0,1,3,5,7,9,11,12}, 70] (* Harvey P. Dale, Jul 06 2014 *)

PARI

concat(0,Vec((1+x)^2*(1-x+x^2)*(1+x+x^2)/(1-x)^2/(1+x+x^2+x^3+x^4+x^5+x^6)+O(x^98))) \\ Charles R Greathouse IV, Mar 11 2012

Formula

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

G.f.: x^2*(1+x)^2*(1-x+x^2)*(1+x+x^2)/((1-x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)). - Colin Barker, Mar 11 2012

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

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

a(n) = (84*n - 84 - 2*(n mod 7) - 2*((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-3, 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)