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

1,3

Comments

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

Links

Table of n, a(n) for n=1..66.

Wikipedia, Neapolitan scale

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

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)

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 *)

Prog

(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
(Magma) [n : n in [0..150] | n mod 12 in [0, 1, 3, 5, 7, 9, 11]]; // Wesley Ivan Hurt, Jul 21 2016

Crossrefs

Cf. A190785.

Sequence in context: A340855 A230078 A275669 * A260392 A186346 A248633

Adjacent sequences: A191272 A191273 A191274 * A191276 A191277 A191278

Keyword

nonn,easy

Author

Roberto Bertocco, May 29 2011

Status

approved