A190440 -id:A190440 - cp's OEIS frontend

A190442 Positions of 1 in A190440.

4, 7, 12, 15, 20, 25, 28, 33, 38, 41, 46, 49, 54, 59, 62, 67, 70, 72, 75, 80, 83, 88, 93, 96, 101, 104, 109, 114, 117, 122, 125, 127, 130, 135, 138, 143, 148, 151, 156, 159, 164, 169, 172, 177, 182, 185, 190, 193, 198, 203, 206, 211, 214, 216, 219, 224, 227, 232, 237, 240, 245, 248, 253, 258, 261, 266, 269, 271, 274, 279, 282, 287, 292, 295, 300

Offset: 1

Clark Kimberling, May 10 2011

nonn

See A190440.

Cf. A190440.

Mathematica
```
(See A190440.)
```

A190443 Positions of 2 in A190440.

Original entry on oeis.org

1, 6, 9, 14, 17, 19, 22, 27, 30, 35, 40, 43, 48, 51, 56, 61, 64, 69, 74, 77, 82, 85, 90, 95, 98, 103, 106, 108, 111, 116, 119, 124, 129, 132, 137, 140, 145, 150, 153, 158, 161, 163, 166, 171, 174, 179, 184, 187, 192, 195, 200, 205, 208, 213, 218, 221, 226, 229, 234, 239, 242, 247, 250, 252, 255, 260, 263, 268, 273, 276, 281, 284, 289, 294, 297

Offset: 1

Clark Kimberling, May 10 2011

nonn

See A190440.

Cf. A190440.

Mathematica
```
(See A190440.)
```

A190248 a(n) = [nu+nv+nw]-[nu]-[nv]-[nw], where u=(1+sqrt(5))/2, v=u^2, w=u^3, []=floor.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 06 2011

nonn

a(n) = A190440(n) - A078588(n). This follows from substituting w = 1+2u, v = 1+u, and taking 2n, n and n out of the floor functions. - Michel Dekking, Oct 21 2016

G. C. Greubel, Table of n, a(n) for n = 1..10000
Burghard Herrmann, How integer sequences find their way into areas outside pure mathematics, The Fibonacci Quarterly (2019) Vol. 57, No. 5, 67-71.

Cf. A190249, A190250, A190251.

[Floor(2*n*(2+Sqrt(5))) - Floor(n*(1+Sqrt(5))/2) - Floor(n*(3 + Sqrt(5))/2): n in [1..30]]; // G. C. Greubel, Dec 26 2017

u = GoldenRatio; v = u^2; w=u^3;
f[n_] := Floor[n*u + n*v + n*w] - Floor[n*u] - Floor[n*v] - Floor[n*w]
t = Table[f[n], {n, 1, 120}] (* A190248 *)
Flatten[Position[t, 0]]      (* A190249 *)
Flatten[Position[t, 1]]      (* A190250 *)
Flatten[Position[t, 2]]      (* A190251 *)

for(n=1,30, print1(floor(2*n*(2+sqrt(5))) - floor(n*(1+sqrt(5))/2) - floor(n*(3 + sqrt(5))/2) - floor(n*(2 + sqrt(5))), ", ")) \\ G. C. Greubel, Dec 26 2017

a(n) = [2n+4nu]-[nu]-[n+nu]-[n+2nu], where u=(1+sqrt(5))/2. - Michel Dekking, Oct 21 2016

Name corrected by Michel Dekking, Oct 21 2016

A190436 a(n) = [(bn+c)r] - b[nr] - [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

Clark Kimberling, May 10 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,2,0):  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

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A190437, A190438, A190439, A190440.

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

Clark Kimberling, May 10 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,2,0):  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

Cf. A190446-A190450, A190427.

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

Clark Kimberling, May 10 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,2,0):  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

Jon Asier Bárcena-Petisco, Luis Martínez, María Merino, Juan Manuel Montoya, and Antonio Vera-López, Fibonacci-like partitions and their associated piecewise-defined permutations, arXiv:2503.19696 [math.CO], 2025. See p. 4.

Cf. A190458-A190461 and A190463.

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]]

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A190442 Positions of 1 in A190440.

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A190443 Positions of 2 in A190440.

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A190248 a(n) = [nu+nv+nw]-[nu]-[nv]-[nw], where u=(1+sqrt(5))/2, v=u^2, w=u^3, []=floor.

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

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

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Mathematica

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

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