A190251 -id:A190251 - cp's OEIS frontend

A190440 [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(golden ratio,4,0) and []=floor.

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

Cf. A190889, A190442, A190443, A190251.

r = GoldenRatio;
f[n_] := Floor[4*n*r] - 4*Floor[n*r];
t = Table[f[n], {n, 1, 320}] (* A190440 *)
Flatten[Position[t, 0]]  (* A190240 *)
Flatten[Position[t, 1]]  (* A190249 *)
Flatten[Position[t, 2]]  (* A190442 *)
Flatten[Position[t, 3]]  (* A190443 *)
Flatten[Position[t, 4]]  (* A190248 *)

a(n)=[4nr]-4[nr], where r=golden ratio.

A190249 Positions of 0 in A190248.

Original entry on oeis.org

2, 5, 10, 13, 18, 23, 26, 31, 34, 36, 39, 44, 47, 52, 57, 60, 65, 68, 73, 78, 81, 86, 89, 91, 94, 99, 102, 107, 112, 115, 120, 123, 128, 133, 136, 141, 146, 149, 154, 157, 162, 167, 170, 175, 178, 180, 183, 188, 191, 196, 201, 204, 209, 212, 217, 222, 225, 230, 233, 235, 238, 243, 246, 251, 256, 259, 264, 267, 272, 277, 280, 285, 290

Offset: 1

Clark Kimberling, May 06 2011

nonn

G. C. Greubel, Table of n, a(n) for n = 1..2700
Burghard Herrmann, How integer sequences find their way into areas outside pure mathematics, The Fibonacci Quarterly (2019) Vol. 57, No. 5, 67-71.

Cf. A190248, A190250, A190251.

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 *)

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

A190250 Positions of 1 in A190248.

Original entry on oeis.org

1, 4, 6, 7, 9, 12, 14, 15, 17, 19, 20, 22, 25, 27, 28, 30, 33, 35, 38, 40, 41, 43, 46, 48, 49, 51, 54, 56, 59, 61, 62, 64, 67, 69, 70, 72, 74, 75, 77, 80, 82, 83, 85, 88, 90, 93, 95, 96, 98, 101, 103, 104, 106, 108, 109, 111, 114, 116, 117, 119, 122, 124, 125, 127, 129, 130, 132, 135, 137, 138, 140, 143, 145, 148, 150, 151, 153, 156, 158, 159, 161

Offset: 1

Clark Kimberling, May 06 2011

nonn

Numbers n such that 1/4 < {n*phi} < 3/4, where phi is the golden ratio (1+sqrt(5))/2 and { } denotes fractional part. - Burghard Herrmann, Nov 14 2017

G. C. Greubel, Table of n, a(n) for n = 1..5500
Burghard Herrmann, Characterization of some golden ratio sequences
Burghard Herrmann, How integer sequences find their way into areas outside pure mathematics, The Fibonacci Quarterly (2019) Vol. 57, No. 5, 67-71.

Cf. A190248, A190249, A190251.

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 *)

isok(n) = my(u=(1+sqrt(5))/2); floor(2*n+4*n*u)-floor(n*u)-floor(n+n*u)-floor(n+2*n*u) == 1; \\ Michel Marcus, Nov 14 2017

A295085 Numbers k such that {kphi} < 0.25 or {kphi} > 0.75, where phi is the golden ratio (1 + sqrt(5))/2 and { } denotes fractional part.

Original entry on oeis.org

2, 3, 5, 8, 10, 11, 13, 16, 18, 21, 23, 24, 26, 29, 31, 32, 34, 36, 37, 39, 42, 44, 45, 47, 50, 52, 53, 55, 57, 58, 60, 63, 65, 66, 68, 71, 73, 76, 78, 79, 81, 84, 86, 87, 89, 91, 92, 94, 97, 99, 100, 102, 105, 107, 110, 112, 113, 115, 118, 120, 121, 123, 126, 128, 131, 133, 134, 136, 139, 141, 142, 144, 146

Offset: 0

Burghard Herrmann, Nov 14 2017

Numbers k such that k rotations by the golden angle yields a result between -Pi/2 and Pi/2 radians.

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Burghard Herrmann, How integer sequences find their way into areas outside pure mathematics, The Fibonacci Quarterly (2019) Vol. 57, No. 5, 67-71.
P. Prusinkiewicz and A. Lindenmayer, Chapter 4, Phyllotaxis, The Algorithmic Beauty of Plants (1990).

Complement of A190250 (as has been proved), thus, intertwining of A190249 and A190251.

Cf. A001622.

Select[Range@ 150, Or[# < 1/4, # > 3/4] &@ FractionalPart[# GoldenRatio] &] (* Michael De Vlieger, Nov 15 2017 *)

isok(n) = my(phi=(1+sqrt(5))/2); (frac(n*phi)<1/4) || (frac(n*phi)>3/4); \\ Michel Marcus, Nov 14 2017

Phi=(sqrt(5)+1)/2 # Golden ratio
fp=function(x) x-floor(x) # fractional part
M=200
alpha=fp((1:M)*Phi) # angles in turn
PF=c(); PB=c() # Phyllotaxis front and back
for (i in 1:M) if ((alpha[i]>0.25)*(alpha[i]<0.75)) PB=c(PB,i) else PF=c(PF,i)

cp's OEIS Frontend

A190440 [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(golden ratio,4,0) and []=floor.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A190249 Positions of 0 in A190248.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A190250 Positions of 1 in A190248.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A295085 Numbers k such that {k*phi} < 0.25 or {k*phi} > 0.75, where phi is the golden ratio (1 + sqrt(5))/2 and { } denotes fractional part.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

R

A295085 Numbers k such that {kphi} < 0.25 or {kphi} > 0.75, where phi is the golden ratio (1 + sqrt(5))/2 and { } denotes fractional part.