A078588 -id:A078588 - cp's OEIS frontend

A089809 Complement of A078588.

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

Offset: 1

Gary W. Adamson, Nov 11 2003

nonn

a(n) = 1 if (fractional part of n*r) < 1/2, else a(n) = 0, where r = golden ratio = (1 + sqrt(5))/2. - Clark Kimberling, Dec 27 2016

			1. a(7) = 1 since A078588(7) = 0
2. a(7) = 1 since A024569 is not 1 (A024569(7) = 3).
3. a(7) = 1 since A089808(7) = 1.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A078588, A024569, A089808.

r = GoldenRatio; z = 500;
Table[If[FractionalPart[n r] < 1/2, 1, 0 ], {n, 1, z}]  (* A089809 *)
Table[If[FractionalPart[n r] > 1/2, 1, 0 ], {n, 1, z}]  (* A078588 *)
1 - % (* A089809,  Clark Kimberling, Dec 27 2016 *)

from math import isqrt
def A089809(n): return ((n+isqrt(5*n**2))&1)^1 # Chai Wah Wu, Aug 17 2022

a(n) = 1 if A078588 = 0; otherwise, not.
a(n) = 1 iff A024569 is not 1.
a(n) = 1 iff A089808 is 1.
a(n) = 1 if (fractional part of n*r) < 1/2, else a(n) = 0. - Clark Kimberling, Dec 27 2016

A082848 Duplicate of A078588.

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

Offset: 0

dead

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.

A005652 Lexicographically least increasing sequence, starting with 1, such that the sum of 2 distinct terms is never a Fibonacci number.

Original entry on oeis.org

1, 3, 6, 8, 9, 11, 14, 16, 17, 19, 21, 22, 24, 27, 29, 30, 32, 35, 37, 40, 42, 43, 45, 48, 50, 51, 53, 55, 56, 58, 61, 63, 64, 66, 69, 71, 74, 76, 77, 79, 82, 84, 85, 87, 90, 92, 95, 97, 98, 100, 103, 105, 106, 108, 110, 111, 113, 116, 118, 119, 121, 124, 126, 129, 131

Offset: 1

N. J. A. Sloane, Simon Plouffe

Also, k such that k = 2*ceiling(k*phi) - ceiling(k*sqrt(5)) where phi = (1+sqrt(5))/2. - Benoit Cloitre, Dec 05 2002
The Chow-Long paper gives a connection with continued fractions, as well as generalizations and other references for this and related sequences.
Positions of 1'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
The lexicographically least property can be proved with the Walnut theorem prover. - Jeffrey Shallit, Nov 20 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].
Primoz Pirnat, Mathematica program

Complement of A005653.
Equals A279934 - 1.
See A078588 for further comments.

f[n_] := Block[{k = Floor[n/GoldenRatio]}, If[n - k*GoldenRatio > (k + 1)*GoldenRatio - n, 1, 0]]; Select[ Range[131], f[ # ] == 1 &]
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]] (* A005653 *)
Flatten[Position[t, 1]] (* this sequence *)
(* 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]] (* A005653 *)
Flatten[Position[t, 1]] (* this sequence *)
(* Clark Kimberling and Jianing Song, Sep 12 2019 *)

The set of all k such that the integer multiple of (1+sqrt(5))/2 nearest k is greater than k (Chow-Long).
Numbers k such that 2*{k*phi} - {2k*phi} = 1, where { } denotes fractional part. - Clark Kimberling, Jan 01 2007
Positive integers k such that A078588(k) = 1. - 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

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

A229829 Numbers coprime to 15.

Original entry on oeis.org

1, 2, 4, 7, 8, 11, 13, 14, 16, 17, 19, 22, 23, 26, 28, 29, 31, 32, 34, 37, 38, 41, 43, 44, 46, 47, 49, 52, 53, 56, 58, 59, 61, 62, 64, 67, 68, 71, 73, 74, 76, 77, 79, 82, 83, 86, 88, 89, 91, 92, 94, 97, 98, 101, 103, 104, 106, 107, 109, 112, 113, 116, 118, 119

Offset: 1

Gary Detlefs, Oct 01 2013

A001651 INTERSECT A047201.
a(n) - 15*floor((n-1)/8) - 2*((n-1) mod 8) has period 8, repeating [1,0,0,1,0,1,1,0].
Numbers whose odd part is 7-rough: products of terms of A007775 and powers of 2 (terms of A000079). - Peter Munn, Aug 04 2020
The asymptotic density of this sequence is 8/15. - Amiram Eldar, Oct 18 2020

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,1,-1).

Lists of numbers coprime to other semiprimes: A007310 (6), A045572 (10), A162699 (14), A160545 (21), A235933 (35).
Cf. A008676, A078588, A113763, A160542, A160543, A160544.
Subsequence of: A001651, A047201.
Subsequences: A000079, A007775.

[n: n in [1..120] | IsOne(GCD(n,15))]; // Bruno Berselli, Oct 01 2013

for n from 1 to 500 do if n mod 3<>0 and n mod 5<>0 then print(n) fi od

Select[Range[120], GCD[#, 15] == 1 &] (* or *) t = 70; CoefficientList[Series[(1 + x + 2 x^2 + 3 x^3 + x^4 + 3 x^5 + 2 x^6 + x^7 + x^8)/((1 - x)^2 (1 + x) (1 + x^2) (1 + x^4)) , {x, 0, t}], x] (* Bruno Berselli, Oct 01 2013 *)
Select[Range[120],CoprimeQ[#,15]&] (* Harvey P. Dale, Oct 31 2013 *)

[i for i in range(120) if gcd(i, 15) == 1] # Bruno Berselli, Oct 01 2013

a(n+8) = a(n) + 15.
a(n) = 15*floor((n-1)/8) +2*f(n) +floor(2*phi*(f(n+1)+2)) -2*floor(phi*(f(n+1)+2)), where f(n) = (n-1) mod 8 and phi=(1+sqrt(5))/2.
a(n) = 15*floor((n-1)/8) +2*f(n) +floor((2*f(n)+5)/5) -floor((f(n)+2)/3), where f(n) = (n-1) mod 8.
From Bruno Berselli, Oct 01 2013: (Start)
G.f.: x*(1 +x +2*x^2 +3*x^3 +x^4 +3*x^5 +2*x^6 +x^7 +x^8) / ((1-x)^2*(1+x)*(1+x^2)*(1+x^4)). -
a(n) = a(n-1) +a(n-8) -a(n-9) for n>9. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*sqrt(7 + sqrt(5) - sqrt(6*(5 + sqrt(5))))*Pi/15. - Amiram Eldar, Dec 13 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.

A024569 [ 1/{n*r} ], where r = (1 + sqrt(5))/2 and {x} := x - [ x ].

Original entry on oeis.org

1, 4, 1, 2, 11, 1, 3, 1, 1, 5, 1, 2, 29, 1, 3, 1, 1, 8, 1, 2, 1, 1, 4, 1, 2, 14, 1, 3, 1, 1, 6, 1, 2, 76, 1, 4, 1, 2, 9, 1, 2, 1, 1, 5, 1, 2, 21, 1, 3, 1, 1, 7, 1, 2, 1, 1, 4, 1, 2, 12, 1, 3, 1, 1, 5, 1, 2, 38, 1, 3, 1, 2, 8, 1, 2, 1, 1, 4, 1, 2, 16, 1, 3, 1, 1, 6, 1, 2, 199, 1, 4, 1, 2, 10

Offset: 1

Clark Kimberling

nonn

A024569(n) = 1 iff A078588 = 1. A024569(n) = 1 iff A089808 is not 1. Either A024569(n) or A089809(n) = 1, but not both. A024569(n) is the complement of A089809(n). - Gary W. Adamson, Nov 11 2003

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A078588, A089808, A089809, A024570.

Table[Floor[1/FractionalPart[n*GoldenRatio]], {n, 100}] (* Clark Kimberling, Aug 15 2012 *)

Corrected by Clark Kimberling, Aug 15 2012

cp's OEIS Frontend

A089809 Complement of A078588.

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A082848 Duplicate of A078588.

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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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A005652 Lexicographically least increasing sequence, starting with 1, such that the sum of 2 distinct terms is never a Fibonacci number.

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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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A229829 Numbers coprime to 15.

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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.