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.

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A190555 [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(sqrt(2),4,2) and []=floor.

Original entry on oeis.org

2, 4, 1, 3, 1, 2, 4, 2, 3, 1, 3, 4, 2, 4, 1, 3, 0, 2, 4, 1, 3, 1, 2, 4, 2, 3, 1, 3, 0, 2, 4, 1, 3, 1, 2, 4, 2, 3, 1, 3, 4, 2, 4, 1, 3, 1, 2, 4, 2, 3, 1, 2, 4, 2, 3, 1, 3, 0, 2, 4, 1, 3, 1, 2, 4, 2, 3, 1, 3, 4, 2, 4, 1, 3, 1, 2, 4, 2, 3, 1, 3, 4, 2, 4, 1, 3, 0, 2, 4, 1, 3, 1, 2, 4, 2, 3, 1, 3, 0, 2, 4, 1, 3, 1, 2, 4, 2, 3, 1, 3, 4, 2, 4, 1, 3, 1, 2, 4, 1, 3, 1, 2, 4, 2, 3, 1, 3, 0
Offset: 1

Views

Author

Clark Kimberling, May 12 2011

Keywords

Comments

Write a(n)=[(bn+c)r]-b[nr]-[cr]. If r>0 and b and c are integers satisfying b>=2 and 0<=c<=b-1, then 0<=a(n)<=b. The positions of 0 in the sequence a are of interest, as are the position sequences for 1,2,...,b. These b+1 position sequences comprise a partition of the positive integers.
Examples:
(golden ratio,2,1): A190427-A190430
(sqrt(2),2,1): A190483-A190486
(sqrt(2),3,0): A190487-A190490
(sqrt(2),3,1): A190491-A190495
(sqrt(2),3,2): A190496-A190500

Crossrefs

Cf. A190556-A190559, A190486.

Programs

  • Mathematica
    r = Sqrt[2]; b = 4; c = 2;
f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
t = Table[f[n], {n, 1, 200}] (* A190555 *)
Flatten[Position[t, 0]]          (* A190556 *)
Flatten[Position[t, 1]]          (* A190557 *)
Flatten[Position[t, 2]]          (* A190558 *)
Flatten[Position[t, 3]]          (* A190559 *)
Flatten[Position[t, 4]]          (* A190486 *)

A190676 [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(sqrt(3),3,0) and [ ]=floor.

Original entry on oeis.org

2, 1, 0, 2, 1, 1, 0, 2, 1, 0, 0, 2, 1, 0, 2, 2, 1, 0, 2, 1, 1, 0, 2, 1, 0, 0, 2, 1, 0, 2, 2, 1, 0, 2, 1, 1, 0, 2, 1, 0, 0, 2, 1, 0, 2, 2, 1, 0, 2, 1, 1, 0, 2, 1, 0, 2, 2, 1, 0, 2, 1, 1, 0, 2, 1, 0, 0, 2, 1, 0, 2, 2, 1, 0, 2, 1, 1, 0, 2, 1, 0, 0, 2, 1, 0, 2, 2, 1, 0, 2, 1, 1, 0, 2, 1, 0, 0, 2, 1, 0, 2, 2, 1, 0, 2, 1, 0, 0, 2, 1, 0, 2, 2, 1, 0, 2, 1, 1, 0, 2, 1, 0, 0, 2, 1, 0, 2, 2, 1, 0, 2, 1
Offset: 1

Views

Author

Clark Kimberling, May 16 2011

Keywords

Comments

Write a(n)=[(bn+c)r]-b[nr]-[cr]. If r>0 and b and c are integers satisfying b>=2 and 0<=c<=b-1, then 0<=a(n)<=b. The positions of 0 in the sequence a are of interest, as are the position sequences for 1,2,...,b. These b+1 position sequences comprise a partition of the positive integers.
Examples:
(golden ratio,2,1): A190427-A190430
(sqrt(2),2,0): A190480-A190482
(sqrt(2),2,1): A190483-A190486
(sqrt(2),3,0): A190487-A190490
(sqrt(2),3,1): A190491-A190495
(sqrt(2),3,2): A190496-A190500
(sqrt(2),4,c): A190544-A190566

Crossrefs

Cf. A190677, A190678, A190679, A022838.

Programs

  • Mathematica
    r = Sqrt[3]; b = 3; c = 0;
f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
t = Table[f[n], {n, 1, 200}] (* A190676 *)
Flatten[Position[t, 0]]      (* A190677 *)
Flatten[Position[t, 1]]      (* A190678 *)
Flatten[Position[t, 2]]      (* A190679 *)
Table[Floor[3n Sqrt[3]]-3Floor[n Sqrt[3]],{n,140}] (* Harvey P. Dale, Mar 24 2013 *)

Formula

a(n)=[3n*sqrt(3)]-3[n*sqrt(3)].

Extensions

Definition (Name) corrected by Harvey P. Dale, Mar 24 2013

A190683 [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(sqrt(3),3,1) and [ ]=floor.

Original entry on oeis.org

2, 2, 1, 3, 2, 1, 1, 3, 2, 1, 0, 3, 2, 1, 3, 2, 2, 1, 3, 2, 1, 1, 3, 2, 1, 0, 3, 2, 1, 3, 2, 2, 1, 3, 2, 1, 0, 3, 2, 1, 0, 2, 2, 1, 3, 2, 1, 1, 3, 2, 1, 0, 3, 2, 1, 3, 2, 2, 1, 3, 2, 1, 1, 3, 2, 1, 0, 3, 2, 1, 3, 2, 2, 1, 3, 2, 1, 1, 3, 2, 1, 0, 3, 2, 1, 3, 2, 1, 1, 3, 2, 1, 0, 3, 2, 1, 0, 2, 2, 1, 3, 2, 1, 1, 3, 2, 1, 0, 3, 2, 1, 3, 2, 2, 1, 3, 2, 1, 1, 3, 2, 1, 0, 3, 2, 1, 3, 2, 2, 1, 3
Offset: 1

Views

Author

Clark Kimberling, May 17 2011

Keywords

Comments

Write a(n)=[(bn+c)r]-b[nr]-[cr]. If r>0 and b and c are integers satisfying b>=2 and 0<=c<=b-1, then 0<=a(n)<=b. The positions of 0 in the sequence a are of interest, as are the position sequences for 1,2,...,b. These b+1 position sequences comprise a partition of the positive integers.
Examples:
(golden ratio,2,1): A190427-A190430
(sqrt(2),2,0): A190480-A190482
(sqrt(2),2,1): A190483-A190486
(sqrt(2),3,0): A190487-A190490
(sqrt(2),3,1): A190491-A190495
(sqrt(2),3,2): A190496-A190500
(sqrt(2),4,c): A190544-A190566

Crossrefs

Cf. A190684-A190687.

Programs

  • Mathematica
    r = Sqrt[3]; b = 3; c = 1;
f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
t = Table[f[n], {n, 1, 200}] (* A190683 *)
Flatten[Position[t, 0]]      (* A190684 *)
Flatten[Position[t, 1]]      (* A190685 *)
Flatten[Position[t, 2]]      (* A190686 *)
Flatten[Position[t, 3]]      (* A190687 *)

A190688 [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(sqrt(3),3,2) and [ ]=floor.

Original entry on oeis.org

2, 1, 1, 3, 2, 1, 0, 3, 2, 1, 0, 2, 2, 1, 3, 2, 1, 0, 3, 2, 1, 0, 2, 2, 1, 0, 2, 1, 1, 3, 2, 1, 0, 3, 2, 1, 0, 2, 2, 1, 0, 2, 1, 1, 3, 2, 1, 0, 3, 2, 1, 0, 2, 2, 1, 3, 2, 1, 1, 3, 2, 1, 0, 3, 2, 1, 0, 2, 1, 1, 3, 2, 1, 0, 3, 2, 1, 0, 2, 2, 1, 0, 2, 1, 1, 3, 2, 1, 0, 3, 2, 1, 0, 2, 2, 1, 0, 2, 1, 1, 3, 2, 1, 0, 3, 2, 1, 0, 2, 2, 1, 3, 2, 1, 1, 3, 2, 1, 0, 3, 2, 1, 0, 2, 1, 1, 3, 2, 1, 0, 3, 2
Offset: 1

Views

Author

Clark Kimberling, May 17 2011

Keywords

Comments

Write a(n)=[(bn+c)r]-b[nr]-[cr]. If r>0 and b and c are integers satisfying b>=2 and 0<=c<=b-1, then 0<=a(n)<=b. The positions of 0 in the sequence a are of interest, as are the position sequences for 1,2,...,b. These b+1 position sequences comprise a partition of the positive integers.
Examples:
(golden ratio,2,1): A190427-A190430
(sqrt(2),2,0): A190480-A190482
(sqrt(2),2,1): A190483-A190486
(sqrt(2),3,0): A190487-A190490
(sqrt(2),3,1): A190491-A190495
(sqrt(2),3,2): A190496-A190500
(sqrt(2),4,c): A190544-A190566

Crossrefs

Cf. A190689-A190692.

Programs

  • Mathematica
    r = Sqrt[3]; b = 3; c = 2;
f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
t = Table[f[n], {n, 1, 200}] (* A190688 *)
Flatten[Position[t, 0]]      (* A190689 *)
Flatten[Position[t, 1]]      (* A190690 *)
Flatten[Position[t, 2]]      (* A190691 *)
Flatten[Position[t, 3]]      (* A190692 *)

A190693 [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(sqrt(3),4,0) and [ ]=floor.

Original entry on oeis.org

2, 1, 0, 3, 2, 1, 0, 3, 2, 1, 0, 3, 2, 0, 3, 2, 1, 0, 3, 2, 1, 0, 3, 2, 1, 0, 3, 1, 0, 3, 2, 1, 0, 3, 2, 1, 0, 3, 2, 1, 0, 2, 1, 0, 3, 2, 1, 0, 3, 2, 1, 0, 3, 2, 1, 3, 2, 1, 0, 3, 2, 1, 0, 3, 2, 1, 0, 3, 2, 0, 3, 2, 1, 0, 3, 2, 1, 0, 3, 2, 1, 0, 3, 1, 0, 3, 2, 1, 0, 3, 2, 1, 0, 3, 2, 1, 0, 2, 1, 0, 3, 2, 1, 0, 3, 2, 1, 0, 3, 2, 1, 3, 2, 1, 0, 3, 2, 1, 0, 3, 2, 1, 0, 3, 2, 0, 3, 2, 1, 0, 3
Offset: 1

Views

Author

Clark Kimberling, May 17 2011

Keywords

Comments

Write a(n)=[(bn+c)r]-b[nr]-[cr]. If r>0 and b and c are integers satisfying b>=2 and 0<=c<=b-1, then 0<=a(n)<=b. The positions of 0 in the sequence a are of interest, as are the position sequences for 1,2,...,b. These b+1 position sequences comprise a partition of the positive integers.
Examples:
(golden ratio,2,1): A190427-A190430
(sqrt(2),2,0): A190480-A190482
(sqrt(2),2,1): A190483-A190486
(sqrt(2),3,0): A190487-A190490
(sqrt(2),3,1): A190491-A190495
(sqrt(2),3,2): A190496-A190500
(sqrt(2),4,c): A190544-A190566

Crossrefs

Cf. A190694-A190697.

Programs

  • Mathematica
    r = Sqrt[3]; b = 4; c = 0;
f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
t = Table[f[n], {n, 1, 200}] (* A190693 *)
Flatten[Position[t, 0]]      (* A190694 *)
Flatten[Position[t, 1]]      (* A190695 *)
Flatten[Position[t, 2]]      (* A190696 *)
Flatten[Position[t, 3]]      (* A190697 *)

Formula

a(n)=[4n*sqrt(3)]-4[n*sqrt(3)].

A190704 [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(sqrt(3),4,2) and [ ]=floor.

Original entry on oeis.org

3, 2, 1, 4, 3, 2, 0, 3, 2, 1, 0, 3, 2, 1, 4, 3, 2, 1, 4, 3, 1, 0, 3, 2, 1, 0, 3, 2, 1, 4, 3, 2, 1, 4, 2, 1, 0, 3, 2, 1, 0, 3, 2, 1, 4, 3, 2, 1, 3, 2, 1, 0, 3, 2, 1, 4, 3, 2, 1, 4, 3, 2, 0, 3, 2, 1, 0, 3, 2, 1, 4, 3, 2, 1, 4, 3, 1, 0, 3, 2, 1, 0, 3, 2, 1, 4, 3, 2, 1, 4, 2, 1, 0, 3, 2, 1, 0, 3, 2, 1, 4, 3, 2, 0, 3, 2, 1, 0, 3, 2, 1, 4, 3, 2, 1, 4, 3, 1, 0, 3, 2, 1, 0, 3, 2, 1, 4, 3, 2, 1, 4, 2
Offset: 1

Views

Author

Clark Kimberling, May 17 2011

Keywords

Comments

Write a(n)=[(bn+c)r]-b[nr]-[cr]. If r>0 and b and c are integers satisfying b>=2 and 0<=c<=b-1, then 0<=a(n)<=b. The positions of 0 in the sequence a are of interest, as are the position sequences for 1,2,...,b. These b+1 position sequences comprise a partition of the positive integers.
Examples:
(golden ratio,2,1): A190427-A190430
(sqrt(2),2,0): A190480-A190482
(sqrt(2),2,1): A190483-A190486
(sqrt(2),3,0): A190487-A190490
(sqrt(2),3,1): A190491-A190495
(sqrt(2),3,2): A190496-A190500
(sqrt(2),4,c): A190544-A190566

Crossrefs

Cf. A190673, A190706-A190709.

Programs

  • Mathematica
    r = Sqrt[3]; b = 4; c = 2;
f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
t = Table[f[n], {n, 1, 200}] (* A190704 *)
Flatten[Position[t, 0]]      (* A190673 *)
Flatten[Position[t, 1]]      (* A190706 *)
Flatten[Position[t, 2]]      (* A190707 *)
Flatten[Position[t, 3]]      (* A190708 *)
Flatten[Position[t, 4]]      (* A190709 *)
With[{r=Sqrt[3],nn=140},Table[Floor[(4n+2)r]-4Floor[n r]-Floor[2r],{n,nn}]] (* Harvey P. Dale, Mar 18 2023 *)

A190770 [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(sqrt(1/2),3,1) and [ ]=floor.

Original entry on oeis.org

2, 1, 1, 3, 2, 1, 3, 2, 1, 0, 3, 2, 1, 3, 2, 1, 0, 2, 2, 1, 3, 2, 1, 3, 2, 1, 0, 3, 2, 1, 3, 2, 1, 0, 2, 2, 1, 3, 2, 1, 3, 2, 1, 1, 3, 2, 1, 3, 2, 1, 0, 3, 2, 1, 3, 2, 1, 0, 2, 1, 1, 3, 2, 1, 3, 2, 1, 0, 3, 2, 1, 3, 2, 1, 0, 2, 2, 1, 3, 2, 1, 3, 2, 1, 1, 3, 2, 1, 3, 2, 1, 0, 2, 2, 1, 3, 2, 1, 0, 2, 1, 1, 3, 2, 1, 3, 2, 1, 0, 3, 2, 1, 3, 2, 1, 0, 2, 2, 1, 3, 2, 1, 3, 2, 1, 0, 3, 2, 1, 3, 2, 1
Offset: 1

Views

Author

Clark Kimberling, May 19 2011

Keywords

Comments

Write a(n)=[(bn+c)r]-b[nr]-[cr]. If r>0 and b and c are integers satisfying b>=2 and 0<=c<=b-1, then 0<=a(n)<=b. The positions of 0 in the sequence a are of interest, as are the position sequences for 1,2,...,b. These b+1 (or b) position sequences comprise a partition of the positive integers.
Examples:
(golden ratio,2,1): A190427-A190430
(sqrt(2),2,0): A190480-A190482
(sqrt(2),2,1): A190483-A190486
(sqrt(2),3,0): A190487-A190490
(sqrt(2),3,1): A190491-A190495
(sqrt(2),3,2): A190496-A190500
(sqrt(2),4,c): A190544-A190566

Crossrefs

Cf. A190771-A190774.

Programs

  • Mathematica
    r = Sqrt[1/2]; b = 3; c = 1;
f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
t = Table[f[n], {n, 1, 200}] (* A190770 *)
Flatten[Position[t, 0]]      (* A190771 *)
Flatten[Position[t, 1]]      (* A190772 *)
Flatten[Position[t, 2]]      (* A190773 *)
Flatten[Position[t, 3]]      (* A190774 *)

A190775 [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(sqrt(1/2),3,2) and [ ]=floor.

Original entry on oeis.org

2, 1, 0, 2, 2, 1, 3, 2, 1, 0, 2, 1, 0, 3, 2, 1, 0, 2, 1, 0, 2, 2, 1, 3, 2, 1, 0, 2, 1, 1, 3, 2, 1, 0, 2, 1, 0, 3, 2, 1, 3, 2, 1, 0, 2, 1, 1, 3, 2, 1, 0, 2, 1, 0, 3, 2, 1, 0, 2, 1, 0, 2, 2, 1, 3, 2, 1, 0, 2, 1, 1, 3, 2, 1, 0, 2, 1, 0, 2, 2, 1, 3, 2, 1, 0, 2, 1, 1, 3, 2, 1, 0, 2, 1, 0, 3, 2, 1, 0, 2, 1, 0, 2, 2, 1, 3, 2, 1, 0, 2, 1, 1, 3, 2, 1, 0, 2, 1, 0, 2, 2, 1, 3, 2, 1, 0, 2, 1, 1, 3, 2, 1
Offset: 1

Views

Author

Clark Kimberling, May 19 2011

Keywords

Comments

Write a(n)=[(bn+c)r]-b[nr]-[cr]. If r>0 and b and c are integers satisfying b>=2 and 0<=c<=b-1, then 0<=a(n)<=b. The positions of 0 in the sequence a are of interest, as are the position sequences for 1,2,...,b. These b+1 (or b) position sequences comprise a partition of the positive integers.
Examples:
(golden ratio,2,1): A190427-A190430
(sqrt(2),2,0): A190480-A190482
(sqrt(2),2,1): A190483-A190486
(sqrt(2),3,0): A190487-A190490
(sqrt(2),3,1): A190491-A190495
(sqrt(2),3,2): A190496-A190500
(sqrt(2),4,c): A190544-A190566

Crossrefs

Cf. A190776-A190779.

Programs

  • Mathematica
    r = Sqrt[1/2]; b = 3; c = 2;
f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
t = Table[f[n], {n, 1, 200}] (* A190775 *)
Flatten[Position[t, 0]]      (* A190776 *)
Flatten[Position[t, 1]]      (* A190777 *)
Flatten[Position[t, 2]]      (* A190778 *)
Flatten[Position[t, 3]]      (* A190779 *)

A190484 Positions of 0 in A190483.

Original entry on oeis.org

3, 5, 10, 15, 17, 20, 22, 27, 29, 32, 34, 39, 44, 46, 51, 56, 58, 61, 63, 68, 73, 75, 80, 85, 87, 90, 92, 97, 99, 102, 104, 109, 114, 116, 119, 121, 126, 128, 131, 133, 138, 143, 145, 150, 155, 157, 160, 162, 167, 169, 172, 174, 179, 184, 186, 189, 191, 196, 198, 201, 203, 208, 213, 215
Offset: 1

Views

Author

Clark Kimberling, May 11 2011

Keywords

Comments

See A190483.

Links

Crossrefs

Cf. A190483.

Programs

  • Mathematica
    r = Sqrt[2]; b = 2; c = 1;
f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
t = Table[f[n], {n, 1, 200}]  (* A190483 *)
Flatten[Position[t, 0]]   (* A190484 *)
Flatten[Position[t, 1]]   (* A190485 *)
Flatten[Position[t, 2]]   (* A190486 *)
  • Python
    from sympy import sqrt, floor
r=sqrt(2)
def a190483(n): return floor((2*n + 1)*r) - 2*floor(n*r) - floor(r)
print([n for n in range(1, 501) if a190483(n)==0]) # Indranil Ghosh, Jul 02 2017

A190485 Positions of 1 in A190483.

Original entry on oeis.org

1, 4, 6, 8, 9, 11, 13, 16, 18, 21, 23, 25, 26, 28, 30, 33, 35, 37, 38, 40, 42, 45, 47, 49, 50, 52, 54, 55, 57, 59, 62, 64, 66, 67, 69, 71, 74, 76, 78, 79, 81, 83, 86, 88, 91, 93, 95, 96, 98, 100, 103, 105, 107, 108, 110, 112, 115, 117, 120, 122, 124, 125, 127, 129, 132, 134, 136, 137, 139, 141, 144, 146, 148, 149, 151, 153, 154, 156, 158
Offset: 1

Views

Author

Clark Kimberling, May 11 2011

Keywords

Comments

See A190483.

Links

Crossrefs

Cf. A190483, A190485, A190486.

Programs

  • Mathematica
    r = Sqrt[2]; b = 2; c = 1;
f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
t = Table[f[n], {n, 1, 200}]  (* A190483 *)
Flatten[Position[t, 0]]   (* A190484 *)
Flatten[Position[t, 1]]   (* A190485 *)
Flatten[Position[t, 2]]   (* A190486 *)
  • Python
    from sympy import sqrt, floor
r=sqrt(2)
def a190483(n): return floor((2*n + 1)*r) - 2*floor(n*r) - floor(r)
print([n for n in range(1, 501) if a190483(n)==1]) # Indranil Ghosh, Jul 02 2017
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