A190485 -id:A190485

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A190483

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

+10
24

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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

(sqrt(2),2,1): A190483-A190486

(sqrt(2),3,0): A190487-A190490

(sqrt(2),3,1): A190491-A190495

(sqrt(2),3,2): A190496-A190500

LINKS

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

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 *)

PROG

(Python)

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

CROSSREFS

Cf. A190484, A190485, A190486.

KEYWORD

nonn

AUTHOR

Clark Kimberling, May 11 2011

STATUS

approved

A190486

Positions of 2 in A190483.

+10
24

2, 7, 12, 14, 19, 24, 31, 36, 41, 43, 48, 53, 60, 65, 70, 72, 77, 82, 84, 89, 94, 101, 106, 111, 113, 118, 123, 130, 135, 140, 142, 147, 152, 159, 164, 171, 176, 181, 183, 188, 193, 200, 205, 210, 212, 217, 222, 229, 234, 239, 241, 246, 251, 253, 258, 263, 270, 275, 280, 282, 287, 292, 299, 304, 309, 311, 316

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

See A190483.

LINKS

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

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 *)

PROG

(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)==2]) # Indranil Ghosh, Jul 02 2017

CROSSREFS

Cf. A190483, A190484, A190485.

KEYWORD

nonn

AUTHOR

Clark Kimberling, May 11 2011

STATUS

approved

A190484

Positions of 0 in A190483.

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

See A190483.

LINKS

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

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 *)

PROG

(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

CROSSREFS

Cf. A190483.

KEYWORD

nonn

AUTHOR

Clark Kimberling, May 11 2011

STATUS

approved

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