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Numbers that are congruent to {0, 1, 3, 5, 7, 8, 11} mod 12.
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%I #41 Sep 08 2022 08:45:57

%S 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,

%T 43,44,47,48,49,51,53,55,56,59,60,61,63,65,67,68,71,72,73,75,77,79,80,

%U 83,84,85,87,89,91,92,95,96,97,99,101,103,104,107,108,109,111

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

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

%C This sequence contains all odd primes. - _Jonathan Vos Post_, Jun 09 2011

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Neapolitan_scale">Neapolitan scale</a>

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,0,1,-1).

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

%F 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

%F a(n) = floor(12*n/7) - floor((n mod 7)/6) - floor(((n+3) mod 7)/5). - _Rolf Pleisch_, Jun 12 2011

%F From _Wesley Ivan Hurt_, Jul 21 2016: (Start)

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

%F 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.

%F 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)

%p 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

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

%o (Magma) [n : n in [0..150] | n mod 12 in [0, 1, 3, 5, 7, 8, 11]]; // _Wesley Ivan Hurt_, Jul 21 2016

%Y Cf. A190785.

%K nonn,easy

%O 1,3

%A _Roberto Bertocco_, May 29 2011