cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: Roberto Bertocco

Roberto Bertocco's wiki page.

Roberto Bertocco has authored 5 sequences.

A349707 Numbers that are congruent to {0, 1, 4, 6, 8, 10, 11} (mod 12).

Original entry on oeis.org

0, 1, 4, 6, 8, 10, 11, 12, 13, 16, 18, 20, 22, 23, 24, 25, 28, 30, 32, 34, 35, 36, 37, 40, 42, 44, 46, 47, 48, 49, 52, 54, 56, 58, 59, 60, 61, 64, 66, 68, 70, 71, 72, 73, 76, 78, 80, 82, 83, 84, 85, 88, 90, 92, 94, 95, 96, 97, 100, 102, 104, 106, 107, 108, 109
Offset: 1

Views

Author

Roberto Bertocco, Nov 26 2021

Keywords

Comments

Terms are the key numbers of the pitches of an Enigmatic scale on a standard chromatic keyboard, with root = 0.

Links

Crossrefs

Cf. A083028.

Programs

  • Mathematica
    upto=200;Select[Range[0,upto],MemberQ[{0,1,4,6,8,10,11},Mod[#,12]]&] (* Paolo Xausa, Nov 30 2021 *)
nterms=100;LinearRecurrence[{1,0,0,0,0,0,1,-1},{0,1,4,6,8,10,11,12},nterms] (* Paolo Xausa, Nov 30 2021 *)
  • Python
    def a(n): return 12*((n-1)//7) + [0, 1, 4, 6, 8, 10, 11][(n-1)%7]
print([a(n) for n in range(1, 66)]) # Michael S. Branicky, Dec 02 2021

Formula

G.f.: x^2*(1 + 3*x + 2*x^2 + 2*x^3 + 2*x^4 + x^5 + x^6)/((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)). - Stefano Spezia, Dec 01 2021
a(n) = a(n-7) + 12 for n >= 8. - Michael S. Branicky, Dec 02 2021

A191276 Numbers that are congruent to {0, 1, 4, 5, 7, 9, 11} mod 12.

Original entry on oeis.org

0, 1, 4, 5, 7, 8, 11, 12, 13, 16, 17, 19, 20, 23, 24, 25, 28, 29, 31, 32, 35, 36, 37, 40, 41, 43, 44, 47, 48, 49, 52, 53, 55, 56, 59, 60, 61, 64, 65, 67, 68, 71, 72, 73, 76, 77, 79, 80, 83, 84, 85, 88, 89, 91, 92, 95, 96, 97, 100, 101, 103, 104, 107, 108, 109, 112, 113, 115, 116, 119, 120, 121, 124, 125, 127, 128, 131, 132, 133, 136, 137, 139, 140, 143, 144, 145, 148, 149, 151, 152, 155, 156, 157, 160, 161, 163, 164, 167, 168, 169, 172, 173, 175, 176, 179, 180, 181
Offset: 1

Views

Author

Roberto Bertocco, May 29 2011

Keywords

Comments

The key-numbers of the pitches of a double harmonic scale (note also as Arabic or Byzantine) on a standard chromatic keyboard, with root = 0.

Links

Crossrefs

Cf. A190785.

Programs

  • Mathematica
    LinearRecurrence[{1,0,0,0,0,0,1,-1},{0,1,4,5,7,8,11,12},120] (* Harvey P. Dale, Mar 24 2019 *)
  • PARI
    concat(0,Vec((1+x+x^2)*(1+2*x-2*x^2+2*x^3+x^4)/(1-x)^2/(1+x+x^2+x^3+x^4+x^5+x^6)+O(x^99))) \\ 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 + x^2)*(1 + 2x - 2x^2 + 2x^3 + x^4)/((1-x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)). - Colin Barker, Mar 11 2012

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

Original entry on oeis.org

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

Views

Author

Roberto Bertocco, May 29 2011

Keywords

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

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)

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

Original entry on oeis.org

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

Views

Author

Roberto Bertocco, May 29 2011

Keywords

Comments

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

Links

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)

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

Original entry on oeis.org

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

Views

Author

Roberto Bertocco, May 26 2011

Keywords

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

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

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