A005652 -id:A005652 - cp's OEIS frontend

A191335 Decimal expansion of sum{(1/3)^A005652(k): k>=1}.

3, 7, 1, 9, 5, 1, 2, 1, 9, 6, 0, 6, 6, 2, 8, 3, 4, 5, 3, 0, 7, 0, 4, 1, 7, 3, 4, 9, 0, 1, 3, 1, 2, 3, 4, 4, 6, 2, 4, 2, 4, 8, 3, 9, 1, 1, 6, 3, 4, 3, 9, 3, 1, 9, 4, 8, 8, 1, 1, 5, 9, 5, 9, 0, 4, 3, 1, 3, 3, 6, 7, 7, 4, 9, 5, 6, 7, 9, 7, 7, 6, 1, 4, 6, 1, 1, 8, 3, 3, 0, 1, 7, 6, 5, 9, 5, 7, 4, 0, 7

Offset: 1

Clark Kimberling, May 31 2011

See A191329 and A191334.

			0.3719512196066283453070417349013123446242...

Cf. A191329, A191334, A005652.

Mathematica
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(See A191329.)
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A183090 Tree generated by A005652, associated with numbers which are not the sum of two Fibonacci numbers.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 8, 7, 11, 12, 9, 10, 16, 15, 14, 13, 21, 23, 22, 25, 17, 18, 19, 20, 30, 33, 29, 31, 27, 28, 24, 26, 42, 41, 45, 46, 43, 44, 50, 49, 32, 34, 35, 36, 37, 38, 40, 39, 58, 60, 64, 67, 56, 59, 61, 62, 53, 54, 55, 57, 48, 47, 51, 52

Offset: 1

Clark Kimberling, Dec 24 2010

A permutation of the positive integers. See the comment at A183079.

			Top 5 rows:
  1;
  2;
  3,             4;
  6,      5,     8,      7;
  11, 12, 9, 10, 16, 15, 14, 13;
From row 3 to row 4: 3->(6,5) and 4->(8,7). For all such pairs, the 1st component is in L and the 2nd, in U.

Cf. A005652, A005653, A183079, A074049.

Let L(n)=A005652(n) and U(n)=A005653(n), these being complementary sequences, each comprising a maximal set no two of whose elements is a Fibonacci number.
The tree-array T(n,k) is then given by rows:
  T(0,0)=1; T(1,0)=2;
  T(n,2*j)=L(T(n-1,j));
  T(n,2*j+1)=U(T(n-1,j));
for j=0,1,...,2^(n-1)-1, n>=2.

A190427 a(n) = [(bn+c)r] - b[nr] - [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

Clark Kimberling, May 10 2011

nonn

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

			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.

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

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

[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

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 *)

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

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

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

A078588 a(n) = 1 if the integer multiple of phi nearest n is greater than n, otherwise 0, where phi = (1+sqrt(5))/2.

Original entry on oeis.org

0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1

Offset: 0

Robert G. Wilson v, Dec 02 2002

From Fred Lunnon, Jun 20 2008: (Start)
Partition the positive integers into two sets A_0 and A_1 defined by A_k == { n | a(n) = k }; so A_0 = A005653 = { 2, 4, 5, 7, 10, 12, 13, 15, 18, 20, ... }, A_1 = A005652 = { 1, 3, 6, 8, 9, 11, 14, 16, 17, 19, 21, ... }.
Then form the sets of sums of pairs of distinct elements from each set and take the complement of their union: this is the Fibonacci numbers { 1, 2, 3, 5, 8, 13, 21, 34, 55, ... } (see the Chow article). (End)
The Chow-Long paper gives a connection with continued fractions, as well as generalizations and other references for this and related sequences.
This is the complement of A089809; also a(n) = 1 iff A024569(n) = 1. - Gary W. Adamson, Nov 11 2003
Since (n*phi) is equidistributed, s(n):=(Sum_{k=1..n}a(k))/n converges to 1/2, but actually s(n) is exactly equal to 1/2 for many values of n. These values are given by A194402. - Michel Dekking, Sep 30 2016
From Clark Kimberling and Jianing Song, Sep 09 2019: (Start)
Suppose that k >= 2, and let a(n) = floor(n*k*r) - k*floor(n*r) = k*{n*r} - {n*k*r}, an integer strictly between 0 and k, where {} denotes fractional part. For h = 0,1,...,k-1, let s(h) be the sequence of positions of h in {a(n)}. The sets s(h) partition the positive integers. Although a(n)/n -> k, the sequence a(n)-k*n appears to be unbounded.
Guide to related sequences, for k = 2:
** r *********  {a(n)}   positions of 0's  positions of 1's
(1+sqrt(5))/2   A078588      A005653           A005652
sqrt(2)         A197879      A120243           A120749
sqrt(3)         A190669      A190670           A190671
e               A190843      A190847           A190860
Pi              A191153      A191159           A191164
sqrt(5)         A188257      A188258           A188259
sqrt(6)         A327253      A327254           A327255
sqrt(8)         A327256      A327257           A327258
Guide to related sequences, for k = 3:
** r *********  {a(n)}   pos. of 0's  pos. of 1's  pos. of 2's
(1+sqrt(5))/2   A140397    A140398      A140399      A140400
sqrt(2)         A190487    A190488      A190489      A190490
sqrt(3)         A190676    A190677      A190678      A190679
e               A190893    A191103      A191104      A191105
Pi              A327298    A327299      A327300      A327301
sqrt(5)         A189463    A189464      A189465      A190158
sqrt(6)         A327306    A327307      A327308      A327309
sqrt(8)         A327310    A327311      A327312      A327313
Guide to related sequences, for k = 4:
** r *********  {a(n)}   pos. of 0's  pos. of 1's  pos. of 2's  pos. of 3's
sqrt(2)         A190544    A190545      A190546      A190547      A190548
sqrt(5)         A189480    A190813      A190883      A190884      A190885
(End)

D. L. Silverman, J. Recr. Math. 9 (4) 208, problem 567 (1976-77).

T. D. Noe, Table of n, a(n) for n = 0..1000
K. Alladi et al., On additive partitions of integers, Discrete Math., 22 (1978), 201-211.
T. Chow, A new characterization of the Fibonacci-free partition, Fibonacci Q. 29 (1991), 174-180; also online here
T. Y. Chow and C. D. Long, Additive partitions and continued fractions, Ramanujan J., 3 (1999), 55-72 [set alpha=(1+sqrt(5))/2 in Theorem 2 to get A005652 and A005653].

Cf. A001622, A005652, A005653, A024569, A089808, A089809, A140397, A140398, A140399, A140400, A140401.

f[n_] := Block[{k = Floor[n/GoldenRatio]}, If[n - k*GoldenRatio > (k + 1)*GoldenRatio - n, 1, 0]]; Table[ f[n], {n, 0, 105}]
r = (1 + Sqrt[5])/2; z = 300;
t = Table[Floor[2 n*r] - 2 Floor[n*r], {n, 0, z}]
(* Clark Kimberling, Aug 26 2019 *)

a(n)=if(n,n+1+ceil(n*sqrt(5))-2*ceil(n*(1+sqrt(5))/2),0) \\ (changed by Jianing Song, Sep 10 2019 to include a(0) = 0)

from math import isqrt
def A078588(n): return (n+isqrt(5*n**2))&1 # Chai Wah Wu, Aug 17 2022

a(n) = floor(2*phi*n) - 2*floor(phi*n) where phi denotes the golden ratio (1 + sqrt(5))/2. - Fred Lunnon, Jun 20 2008
a(n) = 2{n*phi} - {2n*phi}, where { } denotes fractional part. - Clark Kimberling, Jan 01 2007
a(n) = n + 1 + ceiling(n*sqrt(5)) - 2*ceiling(n*phi) where phi = (1+sqrt(5))/2. - Benoit Cloitre, Dec 05 2002
a(n) = round(phi*n) - floor(phi*n). - Michel Dekking, Sep 30 2016
a(n) = (n+floor(n*sqrt(5))) mod 2. - Chai Wah Wu, Aug 17 2022

Edited by N. J. A. Sloane, Jun 20 2008, at the suggestion of Fred Lunnon
Edited by Jianing Song, Sep 09 2019
Offset corrected by Jianing Song, Sep 10 2019

A005653 Lexicographically least increasing sequence, starting with 2, such that the sum of two distinct terms of the sequence is never a Fibonacci number.

Original entry on oeis.org

2, 4, 5, 7, 10, 12, 13, 15, 18, 20, 23, 25, 26, 28, 31, 33, 34, 36, 38, 39, 41, 44, 46, 47, 49, 52, 54, 57, 59, 60, 62, 65, 67, 68, 70, 72, 73, 75, 78, 80, 81, 83, 86, 88, 89, 91, 93, 94, 96, 99, 101, 102, 104, 107, 109, 112, 114, 115, 117, 120, 122, 123, 125, 127, 128

Offset: 1

Simon Plouffe and N. J. A. Sloane

The Chow-Long paper gives a connection with continued fractions, as well as generalizations and other references for this and related sequences.
Positions of 0's in {A078588(n) : n > 0}. - Clark Kimberling and Jianing Song, Sep 10 2019
Also positive integers k such that {k*r} < 1/2, where r = golden ratio = (1 + sqrt(5))/2 and { } = fractional part. - Clark Kimberling and Jianing Song, Sep 12 2019
Jon E. Schoenfield conjectured, and Jeffrey Shallit proved (using the Walnut theorem prover) the characterization in the title. - Jeffrey Shallit, Nov 19 2023

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
K. Alladi et al., On additive partitions of integers, Discrete Math., 22 (1978), 201-211.
T. Y. Chow and C. D. Long, Additive partitions and continued fractions, Ramanujan J., 3 (1999), 55-72 [set alpha=(1+sqrt(5))/2 in Theorem 2 to get A005652 and A005653]. See also on ResearchGate.
Primoz Pirnat, Mathematica program

Complement of A005652. See A078588 for further comments.

f[n_] := Block[{k = Floor[n/GoldenRatio]}, If[n - k*GoldenRatio > (k + 1)*GoldenRatio - n, 1, 0]]; Select[ Range[130], f[ # ] == 0 &]
r = (1 + Sqrt[5])/2; z = 300;
t = Table[Floor[2 n*r] - 2 Floor[n*r], {n, 1, z}] (* {A078588(n) : n > 0} *)
Flatten[Position[t, 0]] (* this sequence *)
Flatten[Position[t, 1]] (* A005652 *)
(* Clark Kimberling and Jianing Song, Sep 10 2019 *)
r = GoldenRatio;
t = Table[If[FractionalPart[n*r] < 1/2, 0, 1 ], {n, 1, 120}] (* {A078588(n) : n > 0} *)
Flatten[Position[t, 0]] (* this sequence *)
Flatten[Position[t, 1]] (* A005652 *)
(* Clark Kimberling and Jianing Song, Sep 12 2019 *)

The set of all n such that the integer multiple of (1+sqrt(5))/2 nearest n is less than n (Chow-Long).
Numbers n such that 2{n*phi}={2n*phi}, where { } denotes fractional part. - Clark Kimberling, Jan 01 2007
Positive integers such that A078588(n) = 0. - Clark Kimberling and Jianing Song, Sep 10 2019

Extended by Robert G. Wilson v, Dec 02 2002

Definition clarified by Jeffrey Shallit, Nov 19 2023

A140397 a(n) = floor(3phin) - 3floor(phin) where phi = (1+sqrt(5))/2.

Original entry on oeis.org

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

Offset: 1

Fred Lunnon, Jun 20 2008

nonn

Cf. A140398, A140399, A140400, A140401, A005652, A005653, A078588.

a[n_]:=Floor[3GoldenRatio*n]-3Floor[GoldenRatio*n];Array[a,99] (* James C. McMahon, Jul 08 2025 *)

a(n) = my(fhi=(1+sqrt(5))/2); floor(3*fhi*n) - 3*floor(fhi*n); \\ Michel Marcus, Aug 26 2013

a(n) = floor(n*s^2)-floor(s*floor(n*s)), where s=1+phi.

A191329 (Lower Wythoff sequence mod 2)+(Upper Wythoff sequence mod 2).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 31 2011

nonn

Let r=(golden ratio)=(1+sqrt(5))/2 and let [ ]=floor. Let u(n)=[nr] and v(n)=n+[nr], so that u=A000201, v=A001950, the Wythoff sequences, and A191329=(u mod 2)+(v mod 2)=(number of odd numbers in {[nr],[ns]}).

The sequence A191329 can also be obtained by placing 1 before each term of 2*A078588.

			u = (1,3,4,6,8,9,...)... = (1,1,0,0,0,1,...) in mod 2
v = (2,5,7,10,13,15,...) = (0,1,1,0,1,1,...) in mod 2,
so that......... A191329 = (1,2,1,0,1,2,...).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000201, A001950, A085002, A171587, A191330, A191331.

r = GoldenRatio; s = r/(r - 1); h = 500;
u = Table[Floor[n*r], {n, 1, h}]  (* A000201 *)
v = Table[Floor[n*s], {n, 1, h}]  (* A001950 *)
w = Mod[u, 2] + Mod[v, 2]  (* A191329 *)
b = Flatten[Position[w, 0]]  (* A191330=2*A005653 *)
c = Flatten[Position[w, 1]]  (* A005408, the odds *)
d = Flatten[Position[w, 2]]  (* A191331=2*A005652 *)
e = b/2; (* A005653 *)
f = d/2; (* A005652 *)
x = (1/3)^b; z = (1/3)^d;
k[n_] := x[[n]]; x1 = Sum[k[n], {n, 1, 100}];
N[x1, 100]
RealDigits[x1, 10, 100]  (* A191332 *)
k[n_] := z[[n]]; z1 = Sum[k[n], {n, 1, 100}];
N[z1, 100]
RealDigits[z1, 10, 100]  (* A191333 *)
N[x1 + z1, 100] (* Checks that x1+z1=1/8 *)
x = (1/3)^e; z = (1/3)^f;
k[n_] := x[[n]]; x2 = Sum[k[n], {n, 1, 100}];
N[x2, 100]
RealDigits[x2, 10, 100]  (* A191334 *)
k[n_] := z[[n]]; z2 = Sum[k[n], {n, 1, 100}];
N[z2, 100]
RealDigits[z2, 10, 100]  (* A191335 *)
N[x2 + z2, 100] (* checks that x2+z2=1/2 *)

A191329(n) = { my(y=n+sqrtint(n^2*5)); (((y+n+n)\2)%2) + ((y%4)>1); }; \\ (after programs in A001950 and A085002) - Antti Karttunen, May 19 2021

from math import isqrt
def A191329(n): return m if (m:=((n+isqrt(5*n**2))&2)+(n&1))<3 else 1 # Chai Wah Wu, Aug 10 2022

a(n) = (A000201(n) mod 2) + (A001950(n) mod 2).

a(n) = A085002(n) + A171587(n). - Michel Dekking, Jan 28 2021

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

Original entry on oeis.org

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.

A114889 a(1)=1 and, for n>1, a(n) is the smallest integer greater than a(n-1) such that a(n)+a(i) is not a power of 3, for i=1,..., n-1.

Original entry on oeis.org

1, 3, 4, 7, 9, 10, 11, 12, 13, 19, 21, 22, 25, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 55, 57, 58, 61, 63, 64, 65, 66, 67, 73, 75, 76, 79, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108

Offset: 1

John W. Layman, Jan 04 2006

nonn

The differences of {a(n)}, together with a conjectured formula for them, is given in A114890.

			Given that a(1)=1, a(2)=3 and a(3)=4, we find that a(4)>5 since 5+4=9 and a(4)>6 since 6+3=9. But none of 7+1, 7+3, or 7+4 is a power of 3, so a(4)=7.

Cf. A005652, A114890.

A190431 a(n) = [(bn+c)r] - b[nr] - [c*r], where (r,b,c)=(golden ratio,3,1) and []=floor.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 10 2011

nonn

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

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

Cf. A190432, A190433, A190434, A190435.

[Floor((3*n+1)*(1+Sqrt(5))/2) - 3*Floor(n*(1+Sqrt(5))/2) - 1: n in [1..100]]; // G. C. Greubel, Apr 06 2018

r = GoldenRatio; b = 3; c = 1;
f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
t = Table[f[n], {n, 1, 320}] (* A190431 *)
Flatten[Position[t, 0]] (* A190432 *)
Flatten[Position[t, 1]] (* A190433 *)
Flatten[Position[t, 2]] (* A190434 *)
Flatten[Position[t, 3]] (* A190435 *)

for(n=1,100, print1(floor((3*n+1)*(1+sqrt(5))/2) - 3*floor(n*(1+sqrt(5))/2) - 1, ", ")) \\ G. C. Greubel, Apr 06 2018

a(n) = floor((3*n+1)*(1+sqrt(5))/2) - 3*floor(n*(1+sqrt(5))/2) - 1. - G. C. Greubel, Apr 06 2018

cp's OEIS Frontend

A191335 Decimal expansion of sum{(1/3)^A005652(k): k>=1}.

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A183090 Tree generated by A005652, associated with numbers which are not the sum of two Fibonacci numbers.

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

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A078588 a(n) = 1 if the integer multiple of phi nearest n is greater than n, otherwise 0, where phi = (1+sqrt(5))/2.

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A005653 Lexicographically least increasing sequence, starting with 2, such that the sum of two distinct terms of the sequence is never a Fibonacci number.

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A140397 a(n) = floor(3*phi*n) - 3*floor(phi*n) where phi = (1+sqrt(5))/2.

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A191329 (Lower Wythoff sequence mod 2)+(Upper Wythoff sequence mod 2).

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

A140397 a(n) = floor(3phin) - 3floor(phin) where phi = (1+sqrt(5))/2.

A190431 a(n) = [(bn+c)r] - b[nr] - [c*r], where (r,b,c)=(golden ratio,3,1) and []=floor.