A005653 -id:A005653 - cp's OEIS frontend

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

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]]

A191330 Positions of 0 in A191329.

Original entry on oeis.org

4, 8, 10, 14, 20, 24, 26, 30, 36, 40, 46, 50, 52, 56, 62, 66, 68, 72, 76, 78, 82, 88, 92, 94, 98, 104, 108, 114, 118, 120, 124, 130, 134, 136, 140, 144, 146, 150, 156, 160, 162, 166, 172, 176, 178, 182, 186, 188, 192, 198, 202, 204, 208, 214, 218, 224, 228, 230, 234, 240, 244, 246, 250, 254, 256, 260, 266, 270, 272, 276, 282, 286

Offset: 1

Clark Kimberling, May 31 2011

nonn

A191330=2*A005653.

			A191329=(1,2,1,0,1,2,1,0,1,0,1,2,1,0,1...),
so that a(1)=4, a(2)=8, a(3)=10,...

Cf. A191329, A005653.

Mathematica
```
(See A191329.)
```

A302253 Positions of 3 in A190436.

Original entry on oeis.org

8, 21, 29, 42, 55, 63, 76, 97, 110, 118, 131, 144, 152, 165, 186, 199, 207, 220, 241, 254, 262, 275, 288, 296, 309, 330, 343, 351, 364, 377, 385, 398, 406, 419, 432, 440, 453, 474, 487, 495, 508, 521, 529, 542, 563, 576, 584, 597, 618, 631, 639, 652, 665, 673, 686, 707, 720, 728

Offset: 1

G. C. Greubel, Apr 04 2018

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..20000

Cf. A190436, A190437, A190438, A190439.

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, 500}] (* A190436 *)
Flatten[Position[t, 0]] (* A190437 *)
Flatten[Position[t, 1]] (* A190438 *)
Flatten[Position[t, 2]] (* A190439 *)
Flatten[Position[t, 3]] (* A302253 *)

A190451 [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(golden ratio,4,2) and []=floor.

Original entry on oeis.org

2, 1, 3, 2, 0, 3, 1, 4, 2, 0, 3, 1, 0, 2, 1, 3, 2, 0, 3, 1, 4, 2, 1, 3, 2, 0, 2, 1, 3, 2, 0, 3, 1, 0, 2, 1, 3, 2, 0, 3, 1, 4, 2, 1, 3, 1, 0, 2, 1, 3, 2, 0, 3, 1, 4, 2, 1, 3, 2, 0, 3, 1, 3, 2, 0, 3, 1, 0, 2, 1, 3, 2, 0, 3, 1, 4, 2, 1, 3, 2, 0, 2, 1, 3, 2, 0, 3, 1, 0, 2, 1, 3, 2, 0, 3, 1, 4, 2, 0, 3, 1, 0, 2, 1, 3, 2, 0, 3, 1, 4, 2, 1, 3, 2, 0, 3, 1, 3, 2, 0, 3, 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

Cf. A190428, A190453-A190455.

r = GoldenRatio; b = 4; 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]]
Flatten[Position[t, 1]]
Flatten[Position[t, 2]]
Flatten[Position[t, 3]]
Flatten[Position[t, 4]]

A140401 Let S be the set of numbers formed from the sum of three distinct elements of A140398, or the sum of three distinct elements of A140399, or the sum of three distinct elements of A140400; sequence gives complement of S.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 18, 21, 23, 26, 29, 31, 34, 39, 42, 47, 55, 60, 68, 76, 81, 89, 102, 110, 123, 144, 157, 178, 199, 212, 233, 267, 288, 322, 377, 411, 466, 521, 555, 610, 699, 754, 843, 987

Offset: 1

Fred Lunnon, Jun 20 2008

Cf. A140398, A140399, A140400, A140397, A005652, A005653, A078588.

It appears that this consists of the following numbers: { F_{k}, F_{k} + F_{k-3}, F_{k} + F_{k-2}, F_{2k} + F_{2k-5}, F_{2k+1} - F_{2k-4}, F_{2k+1} + F_{2k-3} }, where F (A000045) are the Fibonacci numbers and k and other subscripts are restricted to positive values.

A279933 Positive integers k such that {(k-1)*r} < 1/2, where r = golden ratio = (1 + sqrt(5))/2 and { } = fractional part.

Original entry on oeis.org

1, 3, 5, 6, 8, 11, 13, 14, 16, 19, 21, 24, 26, 27, 29, 32, 34, 35, 37, 39, 40, 42, 45, 47, 48, 50, 53, 55, 58, 60, 61, 63, 66, 68, 69, 71, 73, 74, 76, 79, 81, 82, 84, 87, 89, 90, 92, 94, 95, 97, 100, 102, 103, 105, 108, 110, 113, 115, 116, 118

Offset: 1

Clark Kimberling, Dec 23 2016

Clark Kimberling, Table of n, a(n) for n = 1..5000

Cf. A005653, A279934 (complement).

r = GoldenRatio;
t = Table[If[FractionalPart[n r - r] < 1/2, 0, 1 ], {n, 1, 120}] (* {A078588(n-1)} *)
Flatten[Position[t, 0]]  (* A279933 *)
Flatten[Position[t, 1]]  (* A279934 *)

a(n) = 1 + A005653(n-1) for n > 1.

New name from Jianing Song, Sep 12 2019

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.

A367491 Lexicographically least increasing sequence, starting with 2, such that no two terms (possibly identical) sum to a Fibonacci number.

Original entry on oeis.org

2, 5, 7, 9, 10, 13, 15, 18, 20, 22, 23, 26, 28, 30, 31, 34, 36, 38, 39, 41, 43, 44, 47, 49, 52, 54, 56, 57, 60, 62, 64, 65, 68, 70, 73, 75, 77, 78, 81, 83, 85, 86, 89, 91, 93, 94, 96, 98, 99, 102, 104, 107, 109, 111, 112, 115, 117, 119, 120, 123, 125, 127, 128

Offset: 1

Jeffrey Shallit, Nov 20 2023

nonn

There is an 8-state Fibonacci automaton that accepts the Zeckendorf representation of n if and only if n belongs to the sequence.

			6 is not in the sequence, since if it were, 6+2 = 8, a Fibonacci number.

Cf. A005652, A005653. This sequence allows the same term to be used twice in the sum, whereas in the other two sequences, the terms must be distinct.

cp's OEIS Frontend

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

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A191330 Positions of 0 in A191329.

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A302253 Positions of 3 in A190436.

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A190451 [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(golden ratio,4,2) and []=floor.

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A140401 Let S be the set of numbers formed from the sum of three distinct elements of A140398, or the sum of three distinct elements of A140399, or the sum of three distinct elements of A140400; sequence gives complement of S.

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A279933 Positive integers k such that {(k-1)*r} < 1/2, where r = golden ratio = (1 + sqrt(5))/2 and { } = fractional part.

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Mathematica

Formula

Extensions

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

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Comments

Examples

Crossrefs

Formula

A367491 Lexicographically least increasing sequence, starting with 2, such that no two terms (possibly identical) sum to a Fibonacci number.

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Author

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Comments

Examples

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