A190785 Numbers that are congruent to {0, 2, 3, 5, 7, 9, 11} mod 12.

0, 2, 3, 5, 7, 9, 11, 12, 14, 15, 17, 19, 21, 23, 24, 26, 27, 29, 31, 33, 35, 36, 38, 39, 41, 43, 45, 47, 48, 50, 51, 53, 55, 57, 59, 60, 62, 63, 65, 67, 69, 71, 72, 74, 75, 77, 79, 81, 83, 84, 86, 87, 89, 91, 93, 95, 96, 98, 99, 101, 103, 105, 107, 108, 110

Offset: 1

Author

Roberto Bertocco, May 26 2011

Keywords

• nonn
• easy

Comments

The key-numbers of the pitches of a ascending melodic minor scale on a standard chromatic keyboard, with root = 0 and raised seventh.

First differences are period 7: repeat [1,2,2,2,2,1,2]. - Bruno Berselli, May 27 2011

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1,-1).

Crossrefs

Cf. A083028.

Programs

Magma

[n: n in [0..110] | n mod 12 in [0, 2, 3, 5, 7, 9, 11]]; // Bruno Berselli, May 27 2011

Maple

A190785:=n->12*floor(n/7)+[0, 2, 3, 5, 7, 9, 11][(n mod 7)+1]: seq(A190785(n), n=0..100); # Wesley Ivan Hurt, Jul 21 2016

Mathematica

Union[Flatten[Table[12n + {0, 2, 3, 5, 7, 9, 11}, {n, 0, 8}]]] (* Alonso del Arte, Jun 11 2011 *)

PARI

a(n)=n\7*12+[0,2,3,5,7,9,11][n%7+1] \\ Charles R Greathouse IV, Jun 08 2011

Python

def A190785(n):
    a, b = divmod(n-1,7)
    return (0,2,3,5,7,9,11)[b]+12*a # Chai Wah Wu, Jan 26 2023

Formula

a(n) = a(n-1) + a(n-7) - a(n-8) for n>8; G.f.: ( 2+x+2*x^2+2*x^3+2*x^4+2*x^5+x^6 ) / ( (x^6+x^5+x^4+x^3+x^2+x+1)*(x-1)^2 ). - R. J. Mathar, May 26 2011

a(n) = 2*n-floor(2*n/7)-floor(((n-4) mod 7)/5). - Rolf Pleisch, Jun 11 2011

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

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

a(n) = (84*n - 77 - 2*(n mod 7) - 2*((n+1) mod 7) - 2*((n+2) mod 7) - 2*((n+3) mod 7) + 5*((n+4) mod 7) - 2*((n+5) mod 7) + 5*((n+6) mod 7))/49.

a(7*k) = 12*k-1, a(7*k-1) = 12*k-3, a(7*k-2) = 12*k-5, a(7*k-3) = 12*k-7, a(7*k-4) = 12*k-9, a(7*k-5) = 12*k-10, a(7*k-6) = 12*k-12. (End)

Extensions

Zero prepended by Wesley Ivan Hurt, Jul 21 2016