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.

Showing 1-4 of 4 results.

A190427 a(n) = [(b*n+c)*r] - b*[n*r] - [c*r], where (r,b,c)=(golden ratio,2,1) and []=floor.

Original entry on oeis.org

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

Examples

			a(1)=[3r]-2[r]-1=4-3-1=1.
a(2)=[5r]-2[2r]-1=8-6-1=1.
a(3)=[7r]-2[3r]-1=11-8-1=2.

Links

Crossrefs

Cf. A001622, A078588, A190428, A190429, A190430.

Programs

  • Magma
    [Floor((2*n+1)*(1+Sqrt(5))/2) - 2*Floor(n*(1+Sqrt(5))/2) - 1: n in [1..100]]; // G. C. Greubel, Apr 06 2018
  • Mathematica
    r = GoldenRatio; b = 2; c = 1;
f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
t = Table[f[n], {n, 1, 320}] (* A190427 *)
Flatten[Position[t, 0]] (* A190428 *)
Flatten[Position[t, 1]] (* A190429 *)
Flatten[Position[t, 2]] (* A190430 *)
Table[Floor[(2*n+1)*GoldenRatio] - 2*Floor[n*GoldenRatio] -1, {n,1,100}] (* G. C. Greubel, Apr 06 2018 *)
  • PARI
    for(n=1,100, print1(floor((2*n+1)*(1+sqrt(5))/2) - 2*floor(n*(1+sqrt(5))/2) - 1, ", ")) \\ G. C. Greubel, Apr 06 2018
  • Python
    from mpmath import mp, phi
from sympy import floor
mp.dps=100
def a(n): return floor((2*n + 1)*phi) - 2*floor(n*phi) - 1
print([a(n) for n in range(1, 132)]) # Indranil Ghosh, Jul 02 2017

Formula

a(n) = [(2*n+1)*r] - 2*[n*r] - 1, where r=(1+sqrt(5))/2.

A190430 Positions of 2 in A190427.

Original entry on oeis.org

3, 6, 8, 11, 16, 19, 21, 24, 29, 32, 37, 40, 42, 45, 50, 53, 55, 58, 61, 63, 66, 71, 74, 76, 79, 84, 87, 92, 95, 97, 100, 105, 108, 110, 113, 116, 118, 121, 126, 129, 131, 134, 139, 142, 144, 147, 150, 152, 155, 160, 163, 165, 168, 173, 176, 181, 184, 186, 189, 194, 197, 199, 202, 205, 207, 210, 215
Offset: 1

Views

Author

Clark Kimberling, May 10 2011

Keywords

Comments

See A190427.

Links

Crossrefs

Cf. A190427, A190428, A190429.

Programs

  • Mathematica
    r = GoldenRatio; b = 2; c = 1;
f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
t = Table[f[n], {n, 1, 320}] (* A194277 *)
Flatten[Position[t, 0]] (* A190428 *)
Flatten[Position[t, 1]] (* A190429 *)
Flatten[Position[t, 2]] (* A190430 *)

A190429 Positions of 1 in A190427.

Original entry on oeis.org

1, 2, 4, 7, 9, 12, 14, 15, 17, 20, 22, 23, 25, 27, 28, 30, 33, 35, 36, 38, 41, 43, 44, 46, 48, 49, 51, 54, 56, 57, 59, 62, 64, 67, 69, 70, 72, 75, 77, 78, 80, 82, 83, 85, 88, 90, 91, 93, 96, 98, 101, 103, 104, 106, 109, 111, 112, 114, 117, 119, 122, 124, 125, 127, 130, 132, 133, 135, 137, 138, 140, 143, 145, 146, 148, 151, 153, 156, 158
Offset: 1

Views

Author

Clark Kimberling, May 10 2011

Keywords

Comments

See A190427.

Links

Crossrefs

Cf. A190427, A190428, A190430.

Programs

  • Mathematica
    r = GoldenRatio; b = 2; c = 1;
f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
t = Table[f[n], {n, 1, 320}] (* A190427 *)
Flatten[Position[t, 0]] (* A190428 *)
Flatten[Position[t, 1]] (* A190429 *)
Flatten[Position[t, 2]] (* A190430 *)

A190451 [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(golden ratio,4,2) and []=floor.

Original entry on oeis.org

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

Crossrefs

Cf. A190428, A190453-A190455.

Programs

  • Mathematica
    r = GoldenRatio; b = 4; 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]]
Flatten[Position[t, 1]]
Flatten[Position[t, 2]]
Flatten[Position[t, 3]]
Flatten[Position[t, 4]]
Showing 1-4 of 4 results.

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