A190496 -id:A190496 - cp's OEIS frontend

A190500 Positions of 3 in A190496.

2, 7, 9, 12, 14, 19, 24, 26, 31, 36, 38, 41, 43, 48, 53, 55, 60, 65, 67, 70, 72, 77, 82, 84, 89, 94, 96, 101, 106, 108, 111, 113, 118, 123, 125, 130, 135, 137, 140, 142, 147, 152, 154, 159, 164, 166, 171, 176, 178, 181, 183, 188, 193, 195, 200, 205, 207, 210, 212, 217, 222, 224, 229, 234, 236, 239, 241, 246, 248, 251, 253, 258, 263

Offset: 1

Clark Kimberling, May 11 2011

nonn

See A190496.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A190496.

Mathematica
```
(See A190496.)
```

A190498 Positions of 1 in A190496.

Original entry on oeis.org

3, 5, 8, 10, 13, 15, 20, 22, 25, 27, 32, 34, 37, 39, 44, 49, 51, 54, 56, 61, 63, 66, 68, 73, 75, 78, 80, 83, 85, 90, 92, 95, 97, 102, 104, 107, 109, 114, 119, 121, 124, 126, 131, 133, 136, 138, 143, 145, 148, 150, 153, 155, 160, 162, 165, 167, 172, 174, 177, 179, 182, 184, 189, 191, 194, 196, 201, 203, 206, 208, 213, 218, 220, 223

Offset: 1

Clark Kimberling, May 11 2011

nonn

See A190496.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A190496.

Mathematica
```
(See A190496.)
```

A190499 Positions of 2 in A190496.

Original entry on oeis.org

1, 4, 6, 11, 16, 18, 21, 23, 28, 30, 33, 35, 40, 42, 45, 47, 50, 52, 57, 59, 62, 64, 69, 71, 74, 76, 79, 81, 86, 88, 91, 93, 98, 100, 103, 105, 110, 112, 115, 117, 120, 122, 127, 129, 132, 134, 139, 141, 144, 146, 149, 151, 156, 158, 161, 163, 168, 170, 173, 175, 180, 185, 187, 190, 192, 197, 199, 202, 204, 209, 211, 214, 216, 219

Offset: 1

Clark Kimberling, May 11 2011

nonn

See A190496.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A190496.

Mathematica
```
(See A190496.)
```

A190497 Positions of 0 in A190496.

Original entry on oeis.org

17, 29, 46, 58, 87, 99, 116, 128, 157, 169, 186, 198, 215, 227, 256, 268, 285, 297, 326, 338, 355, 367, 396, 425, 437, 454, 466, 495, 507, 524, 536, 565, 577, 594, 606, 623, 635, 664, 676, 693, 705, 734, 746, 763, 775, 792, 804, 833, 845, 862, 874, 903, 915

Offset: 1

Clark Kimberling, May 11 2011

nonn

See A190496.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A190496, A190498, A190499, A190500.

Mathematica
```
(See A190496.)
```

A190487 a(n) = [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(sqrt(2),3,0) and []=floor.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 11 2011

nonn

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
(sqrt(2),2,1):  A190483-A190486
(sqrt(2),3,0):  A190487-A190490
(sqrt(2),3,1):  A190491-A190495
(sqrt(2),3,2):  A190496-A190500

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A190488, A190489, A190490.

r = Sqrt[2]; 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}]  (* A190487 *)
Flatten[Position[t, 0]]   (* A190488 *)
Flatten[Position[t, 1]]   (* A190489 *)
Flatten[Position[t, 2]]   (* A190490 *)

A190491 a(n) = [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(sqrt(2),3,1) and []=floor.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 11 2011

nonn

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, A120743, A170749
(sqrt(2),2,1):  A190483-A190486
(sqrt(2),3,0):  A190487-A190490
(sqrt(2),3,1):  A190491-A190495
(sqrt(2),3,2):  A190496-A190500

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A190492, A190493, A190494, A190495.

r = Sqrt[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}]  (* A190491 *)
Flatten[Position[t, 0]]   (* A190492 *)
Flatten[Position[t, 1]]   (* A190493 *)
Flatten[Position[t, 2]]   (* A190494 *)
Flatten[Position[t, 3]]   (* A190495 *)

A190483 a(n) = [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(sqrt(2),2,1) and []=floor.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 11 2011

nonn

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
(sqrt(2),2,1):  A190483-A190486
(sqrt(2),3,0):  A190487-A190490
(sqrt(2),3,1):  A190491-A190495
(sqrt(2),3,2):  A190496-A190500

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A190484, A190485, A190486.

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 *)

from sympy import sqrt, floor
r=sqrt(2)
def a(n): return floor((2*n + 1)*r) - 2*floor(n*r) - floor(r)
print([a(n) for n in range(1, 501)]) # Indranil Ghosh, Jul 02 2017

A190544 a(n) = [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(sqrt(2),4,0) and []=floor.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 12 2011

nonn

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
(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

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A190545, A190546, A190547, A190548.

r = Sqrt[2]; 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}] (* A190544 *)
Flatten[Position[t, 0]]      (* A190545 *)
Flatten[Position[t, 1]]      (* A190546 *)
Flatten[Position[t, 2]]      (* A190547 *)
Flatten[Position[t, 3]]      (* A190548 *)

a(n) = [4nr] - 4*[nr], where r=sqrt(2).

A190549 a(n) = [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(sqrt(2),4,1) and []=floor.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 12 2011

nonn

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

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A190550, A190551, A190552, A190553.

r = Sqrt[2]; 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}] (* A190549 *)
Flatten[Position[t, 0]]          (* A190550 *)
Flatten[Position[t, 1]]          (* A190551 *)
Flatten[Position[t, 2]]          (* A190552 *)
Flatten[Position[t, 3]]          (* A190553 *)
Flatten[Position[t, 4]]          (* A190554 *)

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

Clark Kimberling, May 12 2011

nonn

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

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A190562 to A190566.

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 *)

cp's OEIS Frontend

A190500 Positions of 3 in A190496.

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A190498 Positions of 1 in A190496.

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A190499 Positions of 2 in A190496.

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A190497 Positions of 0 in A190496.

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A190487 a(n) = [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(sqrt(2),3,0) and []=floor.

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A190491 a(n) = [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(sqrt(2),3,1) and []=floor.

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A190483 a(n) = [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(sqrt(2),2,1) and []=floor.

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A190544 a(n) = [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(sqrt(2),4,0) and []=floor.

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A190549 a(n) = [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(sqrt(2),4,1) and []=floor.

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