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.

Previous Showing 11-20 of 31 results. Next

A190561 a(n) = [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(sqrt(2),4,3) and []=floor.

Original entry on oeis.org

1, 3, 1, 2, 0, 2, 3, 1, 3, 0, 2, 4, 1, 3, 1, 2, 0, 2, 3, 1, 3, 0, 2, 4, 1, 3, 0, 2, 0, 1, 3, 1, 2, 0, 2, 3, 1, 3, 0, 2, 4, 1, 3, 1, 2, 0, 2, 3, 1, 3, 0, 2, 4, 1, 3, 1, 2, 0, 1, 3, 1, 2, 0, 2, 3, 1, 3, 0, 2, 4, 1, 3, 1, 2, 0, 2, 3, 1, 3, 0, 2, 4, 1, 3, 1, 2, 0, 2, 3, 1, 3, 0, 2, 3, 1, 3, 0, 2, 0, 1, 3, 1, 2, 0, 2, 3, 1, 3, 0, 2, 4, 1, 3, 1, 2, 0, 2, 3, 1, 3, 0, 2, 4, 1, 3, 1, 2, 0, 1, 3, 1, 2, 0, 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),2,0): 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

Programs

  • Mathematica
    r = Sqrt[2]; b = 4; c = 3;
    f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
    t = Table[f[n], {n, 1, 200}] (* A190561 *)
    Flatten[Position[t, 0]]          (* A190562 *)
    Flatten[Position[t, 1]]          (* A190563 *)
    Flatten[Position[t, 2]]          (* A190564 *)
    Flatten[Position[t, 3]]          (* A190565 *)
    Flatten[Position[t, 4]]          (* A190566 *)

A190698 a(n) = [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(sqrt(3),4,1) and [ ]=floor.

Original entry on oeis.org

3, 2, 1, 4, 3, 2, 1, 4, 3, 2, 0, 3, 2, 1, 4, 3, 2, 1, 4, 3, 2, 1, 4, 3, 1, 0, 3, 2, 1, 4, 3, 2, 1, 4, 3, 2, 1, 4, 2, 1, 0, 3, 2, 1, 4, 3, 2, 1, 4, 3, 2, 0, 3, 2, 1, 4, 3, 2, 1, 4, 3, 2, 1, 4, 3, 1, 0, 3, 2, 1, 4, 3, 2, 1, 4, 3, 2, 1, 4, 2, 1, 0, 3, 2, 1, 4, 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, 4, 3, 2, 1, 4, 3, 2, 1, 4, 3, 1, 0, 3, 2, 1, 4
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

Programs

  • Mathematica
    r = Sqrt[3]; b = 4; c = 1;
    f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
    t = Table[f[n], {n, 1, 200}] (* A190698 *)
    Flatten[Position[t, 0]]      (* A190699 *)
    Flatten[Position[t, 1]]      (* A190700 *)
    Flatten[Position[t, 2]]      (* A190701 *)
    Flatten[Position[t, 3]]      (* A190702 *)
    Flatten[Position[t, 4]]      (* A190703 *)

A190710 [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(sqrt(3),4,3) and [ ]=floor.

Original entry on oeis.org

3, 2, 0, 3, 2, 1, 0, 3, 2, 1, 0, 3, 2, 1, 4, 3, 1, 0, 3, 2, 1, 0, 3, 2, 1, 0, 3, 2, 1, 4, 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, 4, 3, 2, 0, 3, 2, 1, 0, 3, 2, 1, 0, 3, 2, 1, 4, 3, 1, 0, 3, 2, 1, 0, 3, 2, 1, 0, 3, 2, 1, 4, 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, 4, 3, 2, 0, 3, 2, 1, 0, 3, 2, 1, 0, 3, 2, 1, 4, 3, 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

Programs

  • Mathematica
    r = Sqrt[3]; b = 4; c = 3;
    f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
    t = Table[f[n], {n, 1, 200}] (* A190710 *)
    Flatten[Position[t, 0]]      (* A190711 *)
    Flatten[Position[t, 1]]      (* A190712 *)
    Flatten[Position[t, 2]]      (* A190713 *)
    Flatten[Position[t, 3]]      (* A190714 *)
    Flatten[Position[t, 4]]      (* A190715 *)

A190762 [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(sqrt(1/2),2,1) and [ ]=floor.

Original entry on oeis.org

2, 1, 0, 2, 1, 1, 2, 2, 1, 0, 2, 1, 1, 2, 1, 1, 0, 2, 1, 0, 2, 1, 1, 2, 2, 1, 0, 2, 1, 1, 2, 1, 1, 0, 2, 1, 1, 2, 1, 1, 2, 2, 1, 0, 2, 1, 1, 2, 2, 1, 0, 2, 1, 1, 2, 1, 1, 0, 2, 1, 0, 2, 1, 1, 2, 2, 1, 0, 2, 1, 1, 2, 1, 1, 0, 2, 1, 1, 2, 1, 1, 2, 2, 1, 0, 2, 1, 1, 2, 1, 1, 0, 2, 1, 1, 2, 1, 1, 0, 2, 1, 0, 2, 1, 1, 2, 2, 1, 0, 2, 1, 1, 2, 1, 1, 0, 2, 1, 0, 2, 1, 1, 2, 2, 1, 0, 2, 1, 1, 2, 1, 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 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

Programs

  • Mathematica
    r = Sqrt[1/2]; 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}] (* A190762 *)
    Flatten[Position[t, 0]]      (* A190763 *)
    Flatten[Position[t, 1]]      (* A190764 *)
    Flatten[Position[t, 2]]      (* A190765 *)

A190431 a(n) = [(b*n+c)*r] - b*[n*r] - [c*r], where (r,b,c)=(golden ratio,3,1) and []=floor.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, May 10 2011

Keywords

Comments

Write a(n) = [(b*n+c)*r] - b*[n*r] - [c*r]. 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,2,0): A078588, A005653, A005652
(golden ratio,2,1): A190427 - A190430
(golden ratio,3,0): A140397 - A190400
(golden ratio,3,1): A140431 - A190435
(golden ratio,3,2): A140436 - A190439

Crossrefs

Programs

  • Magma
    [Floor((3*n+1)*(1+Sqrt(5))/2) - 3*Floor(n*(1+Sqrt(5))/2) - 1: n in [1..100]]; // G. C. Greubel, Apr 06 2018
  • Mathematica
    r = GoldenRatio; b = 3; c = 1;
    f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
    t = Table[f[n], {n, 1, 320}] (* A190431 *)
    Flatten[Position[t, 0]] (* A190432 *)
    Flatten[Position[t, 1]] (* A190433 *)
    Flatten[Position[t, 2]] (* A190434 *)
    Flatten[Position[t, 3]] (* A190435 *)
  • PARI
    for(n=1,100, print1(floor((3*n+1)*(1+sqrt(5))/2) - 3*floor(n*(1+sqrt(5))/2) - 1, ", ")) \\ G. C. Greubel, Apr 06 2018
    

Formula

a(n) = floor((3*n+1)*(1+sqrt(5))/2) - 3*floor(n*(1+sqrt(5))/2) - 1. - G. C. Greubel, Apr 06 2018

A190436 a(n) = [(b*n+c)*r] - b*[n*r] - [c*r], where (r,b,c)=(golden ratio,3,2) and []=floor.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, May 10 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,2,0): A078588, A005653, A005652
(golden ratio,2,1): A190427 - A190430
(golden ratio,3,0): A140397 - A190400
(golden ratio,3,1): A140431 - A190435
(golden ratio,3,2): A140436 - A190439
(golden ratio,4,c): A140440 - A190461

Crossrefs

Programs

  • Mathematica
    r = GoldenRatio; b = 3; c = 2;
    f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
    t = Table[f[n], {n, 1, 320}]
    Flatten[Position[t, 0]] (* A190437 *)
    Flatten[Position[t, 1]] (* A190438 *)
    Flatten[Position[t, 2]] (* A190439 *)
    Flatten[Position[t, 3]] (* A302253 *)

A190445 [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(golden ratio,4,1) and []=floor.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, May 10 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,2,0): A078588, A005653, A005652
(golden ratio,2,1): A190427-A190430
(golden ratio,3,0): A140397-A190400
(golden ratio,3,1): A140431-A190435
(golden ratio,3,2): A140436-A190439
(golden ratio,4,c): A190440-A190461

Crossrefs

Programs

  • Mathematica
    r = GoldenRatio; b = 4; c = 1;
    f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
    t = Table[f[n], {n, 1, 320}]
    Flatten[Position[t, 0]]
    Flatten[Position[t, 1]]
    Flatten[Position[t, 2]]
    Flatten[Position[t, 3]]
    Flatten[Position[t, 4]]

A190457 a(n) = [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(golden ratio,4,3) and []=floor.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, May 10 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,2,0): A078588, A005653, A005652
(golden ratio,2,1): A190427-A190430
(golden ratio,3,0): A140397-A190400
(golden ratio,3,1): A140431-A190435
(golden ratio,3,2): A140436-A190439
(golden ratio,4,c): A190440-A190461

Crossrefs

Programs

  • Mathematica
    r = GoldenRatio; b = 4; c = 3;
    f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
    t = Table[f[n], {n, 1, 320}]
    Flatten[Position[t, 0]]
    Flatten[Position[t, 1]]
    Flatten[Position[t, 2]]
    Flatten[Position[t, 3]]
    Flatten[Position[t, 4]]

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

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

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
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