A026355 -id:A026355 - cp's OEIS frontend

A007067 Nearest integer to n*tau where tau = (1+sqrt(5))/2.

0, 2, 3, 5, 6, 8, 10, 11, 13, 15, 16, 18, 19, 21, 23, 24, 26, 28, 29, 31, 32, 34, 36, 37, 39, 40, 42, 44, 45, 47, 49, 50, 52, 53, 55, 57, 58, 60, 61, 63, 65, 66, 68, 70, 71, 73, 74, 76, 78, 79, 81, 83, 84, 86, 87, 89, 91, 92, 94, 95, 97, 99, 100, 102, 104, 105

Offset: 0

N. J. A. Sloane, Mira Bernstein

First column of inverse Stolarsky array.

A rectangle of size a(n) X n approximates a golden rectangle. So does A295282(n) X n, which targets the golden ratio's underlying objective. These approximations differ first for n = 4 and generally if n = F(6*k)/2, where F(n) = A000045(n) is the n-th Fibonacci number and k >= 1. - Peter Munn, Jan 12 2018

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

Iain Fox, Table of n, a(n) for n = 0..10000 (first 1001 terms from Vincenzo Librandi)
Clark Kimberling, Interspersions and dispersions, Proceedings of the American Mathematical Society, 117 (1993) 313-321.
Luke Schaeffer, Jeffrey Shallit, and Stefan Zorcic, Beatty Sequences for a Quadratic Irrational: Decidability and Applications, arXiv:2402.08331 [math.NT], 2024.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
N. J. A. Sloane, Classic Sequences.

Cf. A166946 (characteristic function), A007064 (complement).
Different from A026355.
Sequences with similar terms: A022342, A295282.
Other roundings of n*tau: A000201, A004956, A066096.
Cf. A000045 (Fibonacci numbers), A001622 (value of tau).

A007067:=n->round(n*(1+sqrt(5))/2); seq(A007067(n), n=0..100); # Wesley Ivan Hurt, Nov 27 2013

a[n_] := Round[n*GoldenRatio]; Table[a[n], {n, 0, 55}] (* Jean-François Alcover, Jun 27 2012 *)

a(n) = round(n*(1+sqrt(5))/2) \\ Michel Marcus, May 20 2013

from math import isqrt
def A007067(n): return (isqrt(5*n**2<<2)>>1)+n+1>>1 # Chai Wah Wu, Aug 26 2022

Satisfies a(a(n)) = a(n) + n. - Franklin T. Adams-Watters, Aug 14 2006

a(n) = floor((A066096(2*n) + 1)/2). - Peter Munn, Jan 12 2018

A007066 a(n) = 1 + ceiling((n-1)*phi^2), phi = (1+sqrt(5))/2.

Original entry on oeis.org

1, 4, 7, 9, 12, 15, 17, 20, 22, 25, 28, 30, 33, 36, 38, 41, 43, 46, 49, 51, 54, 56, 59, 62, 64, 67, 70, 72, 75, 77, 80, 83, 85, 88, 91, 93, 96, 98, 101, 104, 106, 109, 111, 114, 117, 119, 122, 125, 127, 130, 132, 135, 138, 140, 143, 145, 148, 151, 153, 156, 159, 161, 164, 166

Offset: 1

N. J. A. Sloane, Mira Bernstein

First column of dual Wythoff array, A126714.
Positions of 0's in A189479.
Skala (2016) asks if this sequence also gives the positions of the 0's in A283310. - N. J. A. Sloane, Mar 06 2017
Upper Wythoff sequence plus 2, when shifted by 1. - Michel Dekking, Aug 26 2019
In the Fokkink-Joshi paper, this sequence is the Cloitre (0,1,2,3)-hiccup sequence, i.e., a(1) = 1; for m < n, a(n) = a(n-1)+2 if a(m) = n, else a(n) = a(n-1)+3. - Michael De Vlieger, Jul 30 2025

Clark Kimberling, "Stolarsky interspersions," Ars Combinatoria 39 (1995) 129-138.
D. R. Morrison, "A Stolarsky array of Wythoff pairs," in A Collection of Manuscripts Related to the Fibonacci Sequence. Fibonacci Assoc., Santa Clara, CA, 1980, pp. 134-136.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Benoit Cloitre, A study of a family of self-referential sequences, arXiv:2506.18103 [math.GM], 2025. See p. 9.
Robbert Fokkink and Gandhar Joshi, On Cloitre's hiccup sequences, arXiv:2507.16956 [math.CO], 2025. See pp. 3-4, 7, 10.
Clark Kimberling, Interspersions
Matthew Skala, Graph Nimors, arXiv preprint arXiv:1604.04072 [math.CO], 2016.
N. J. A. Sloane, Classic Sequences

Cf. A064437.
Apart from initial terms, same as A026356 (Cloitre (0,2,2,3)-hiccup sequence).
First column of A126714.
Complement is (essentially) A026355.
Equals 1 + A004957, also n + A004956.
First differences give A076662.
Complement of A099267. [Gerald Hillier, Dec 19 2008]
Cf. A193214 (primes). Except for the first term equal to A001950 + 2.
Cf. A026352 (Cloitre (1,1,2,3)-hiccup sequence), A064437 (Cloitre (0,1,3,2)-hiccup sequence).

a007066 n = a007066_list !! (n-1)
a007066_list = 1 : f 2 [1] where
   f x zs@(z:_) = y : f (x + 1) (y : zs) where
     y = if x `elem` zs then z + 2 else z + 3
-- Reinhard Zumkeller, Sep 26 2014, Sep 18 2011

Digits := 100: t := (1+sqrt(5))/2; A007066 := proc(n) if n <= 1 then 1 else floor(1+t*floor(t*(n-1)+1)); fi; end;

t = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {1, 0, 1}}] &, {0}, 6] (*A189479*)
Flatten[Position[t, 0]] (*A007066*)
Flatten[Position[t, 1]] (*A099267*)
With[{grs=GoldenRatio^2},Table[1+Ceiling[grs(n-1)],{n,70}]] (* Harvey P. Dale, Jun 24 2011 *)

from math import isqrt
def A007066(n): return (n+1+isqrt(5*(n-1)**2)>>1)+n if n > 1 else 1 # Chai Wah Wu, Aug 25 2022

a(n) = floor(1+phi*floor(phi*(n-1)+1)), phi=(1+sqrt(5))/2, n >= 2.
a(1)=1; for n>1, a(n)=a(n-1)+2 if n is already in the sequence, a(n)=a(n-1)+3 otherwise. - Benoit Cloitre, Mar 06 2003
a(n+1) = floor(n*phi^2) + 2, n>=1. - Michel Dekking, Aug 26 2019

A099267 Numbers generated by the golden sieve.

Original entry on oeis.org

2, 3, 5, 6, 8, 10, 11, 13, 14, 16, 18, 19, 21, 23, 24, 26, 27, 29, 31, 32, 34, 35, 37, 39, 40, 42, 44, 45, 47, 48, 50, 52, 53, 55, 57, 58, 60, 61, 63, 65, 66, 68, 69, 71, 73, 74, 76, 78, 79, 81, 82, 84, 86, 87, 89, 90, 92, 94, 95, 97, 99, 100, 102, 103, 105, 107, 108, 110

Offset: 1

Benoit Cloitre, Nov 15 2002

Let f(n) denote the n-th term of the current working sequence. Start with the positive integers:
1,2,3,4,5,6,7,8,9,10,11,12,...
Delete the term in position f(1), which is f(f(1))=f(1)=1, leaving:
2,3,4,5,6,7,8,9,10,11,12,...
Delete the term in position f(2), which is f(f(2))=f(3)=4, leaving:
2,3,5,6,7,8,9,10,11,12,...
Delete the term in position f(3), which is f(f(3))=f(5)=7, leaving:
2,3,5,6,8,9,10,11,12,...
Delete the term in position f(4), which is f(f(4))=f(6)=9, leaving:
2,3,5,6,8,10,11,12,...
Iterating the "sieve" indefinitely produces the sequence:
2,3,5,6,8,10,11,13,14,16,18,19,21,23,24,26,27,29,31,32,34,35,37,39,...
Positions of 1 in A189479. - Clark Kimberling, Apr 22 2011

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
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. 3.
Index entries for sequences generated by sieves

Numbers n such that a(n+1)-a(n)=2 are given by A004956.
If prefixed by an initial 1, same as A026355.
Cf. A001622, A136119, A007066, A189479.
Complement of A007066. - Gerald Hillier, Dec 19 2008
Cf. A193213 (primes).

a099267 n = a099267_list !! (n-1)
a099267_list = f 1 [1..] 0 where
   f k xs y = ys' ++ f (k+1) (ys ++ xs') g where
     ys' = dropWhile (< y) ys
     (ys,_:xs') = span (< g) xs
     g = xs !! (h - 1)
     h = xs !! (k - 1)
-- Reinhard Zumkeller, Sep 18 2011

t = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {1, 0, 1}}] &, {0}, 6] (*A189479*)
Flatten[Position[t, 0]] (*A007066*)
Flatten[Position[t, 1]] (*A099267*)

a(n) = floor(n*phi + 2 - phi) where phi = (1 + sqrt(5))/2.
a(a(...a(1)...)) with n iterations equals F(n+1) = A000045(n+1).
For n>0 and k>0 we have a(a(n) + F(k) - (1 + (-1)^k)/2) = a(a(n)) + F(k+1) - 1 - (-1)^k. - Benoit Cloitre, Nov 22 2004
a(n) = a(a(n)) - n. - Marc Morgenegg, Sep 23 2019

A026356 a(n) = floor((n-1)*phi) + n + 1, n > 0, where phi = (1+sqrt(5))/2.

Original entry on oeis.org

2, 4, 7, 9, 12, 15, 17, 20, 22, 25, 28, 30, 33, 36, 38, 41, 43, 46, 49, 51, 54, 56, 59, 62, 64, 67, 70, 72, 75, 77, 80, 83, 85, 88, 91, 93, 96, 98, 101, 104, 106, 109, 111, 114, 117, 119, 122, 125, 127, 130, 132, 135, 138, 140, 143, 145

Offset: 1

Clark Kimberling

nonn

Greatest k such that s(k) = n+1, where s = A026354.
Positions of 1 in A189661.
a(n+1) = A001950(n)-2, the Upper Wythoff sequence shifted by 2. - Michel Dekking, Oct 18 2018
In the Fokkink-Joshi paper, this sequence is the Cloitre (0,2,2,3)-hiccup sequence, i.e., a(1) = 2; for m < n, a(n) = a(n-1)+2 if a(m) = n, else a(n) = a(n-1)+3. - Michael De Vlieger, Jul 28 2025

Michel Dekking, Table of n, a(n) for n = 1..1000
Benoit Cloitre, A study of a family of self-referential sequences, arXiv:2506.18103 [math.GM], 2025. See p. 9.
Robbert Fokkink and Gandhar Joshi, On Cloitre's hiccup sequences, arXiv:2507.16956 [math.CO], 2025. See pp. 7, 10, 17.

Cf. A000201, A026351, etc. Apart from initial terms, same as A007066. Complement is A189662, closely related to A026355.

(* See A189661. Second program: *)
cloitreH[j_, x_, y_, z_, w_ : 120] := Block[{c, k}, c[] := False; k = x; c[x] = True; {x}~Join~Reap[Do[If[c[n - j], k += y, k += z]; c[k] = True; Sow[k], {n, 2, w}] ][[-1, 1]] ]; cloitreH[0, 2, 2, 3] (* _Michael De Vlieger, Jul 28 2025 *)

r = (1 + sqrt(5))/2;
a(n) = if(n<1, 1, floor((n - 1)* r) + n + 1);
for(n=1, 100, print1(a(n),", ")) \\ Indranil Ghosh, Mar 25 2017

from sympy import sqrt
import math
r=(1 + sqrt(5))/2
def a(n): return 1 if n<1 else int(math.floor((n - 1)*r)) + n + 1
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Mar 25 2017

from math import isqrt
def A026356(n): return (n+1+isqrt(5*(n-1)**2)>>1)+n # Chai Wah Wu, Aug 11 2022

Data corrected by Michel Dekking, Oct 18 2018

A189662 Positions of 0 in A189661; complement of A026356.

Original entry on oeis.org

1, 3, 5, 6, 8, 10, 11, 13, 14, 16, 18, 19, 21, 23, 24, 26, 27, 29, 31, 32, 34, 35, 37, 39, 40, 42, 44, 45, 47, 48, 50, 52, 53, 55, 57, 58, 60, 61, 63, 65, 66, 68, 69, 71, 73, 74, 76, 78, 79, 81, 82, 84, 86, 87, 89, 90, 92, 94, 95, 97, 99, 100, 102, 103, 105, 107, 108, 110, 112, 113, 115, 116, 118, 120, 121, 123, 124, 126, 128, 129, 131, 133, 134, 136, 137, 139, 141, 142, 144

Offset: 1

Clark Kimberling, Apr 25 2011

nonn

See A189661.

Apparently the same as A099267 from the 2nd entry on, and in consequence essentially also A026355. - R. J. Mathar, May 16 2011

Cf. A189661, A026356, A189663.

Cf. A026355, A099267.

Mathematica
```
(See A189661.)
```

A054082 Permutation of N: a(1)=2, a(2)=1 and for each k >= 2, let p(k)=least natural number not already an a(i), q(k)=p(k)+k-1, a(2k-1)=q(k), a(2k)=p(k).

Original entry on oeis.org

2, 1, 4, 3, 7, 5, 9, 6, 12, 8, 15, 10, 17, 11, 20, 13, 22, 14, 25, 16, 28, 18, 30, 19, 33, 21, 36, 23, 38, 24, 41, 26, 43, 27, 46, 29, 49, 31, 51, 32, 54, 34, 56, 35, 59, 37, 62, 39, 64, 40, 67, 42, 70, 44, 72, 45, 75, 47, 77, 48, 80, 50, 83

Offset: 1

Clark Kimberling

Index entries for sequences that are permutations of the natural numbers

Odd-indexed terms: A026356. Even-indexed terms: A026355. Inverse permutation: A064579.

a[n_] := If[OddQ[n], Floor[((n+1)/2 - 1) GoldenRatio] + (n+1)/2 + 1, Floor[(n/2 - 1) GoldenRatio] + 2]; a[2] = 1;
Array[a, 100] (* Jean-François Alcover, Apr 01 2020 *)

A332502 Rectangular array read by antidiagonals: T(n,k) = floor(n + k*r), where r = golden ratio = (1+sqrt(5))/2.

Original entry on oeis.org

0, 1, 1, 3, 2, 2, 4, 4, 3, 3, 6, 5, 5, 4, 4, 8, 7, 6, 6, 5, 5, 9, 9, 8, 7, 7, 6, 6, 11, 10, 10, 9, 8, 8, 7, 7, 12, 12, 11, 11, 10, 9, 9, 8, 8, 14, 13, 13, 12, 12, 11, 10, 10, 9, 9, 16, 15, 14, 14, 13, 13, 12, 11, 11, 10, 10, 17, 17, 16, 15, 15, 14, 14, 13

Offset: 0

Clark Kimberling, May 08 2020

Every nonnegative integer occurs exactly once in the union of row 0 and the main diagonal.
Column 0: Nonnegative integers, A001477.
Row 0: Lower Wythoff sequence, A000201.
Row 1: A026351.
Row 2: A026355 (and essentially A099267).
Main Diagonal: Upper Wythoff sequence, A001950.
Diagonal (1,4,6,9,...) = A003622;
Diagonal (3,5,8,11,...) = A026274;
Diagonal (1,3,6,8,...) = A026352;
Diagonal (2,4,7,9,...) = A026356.

			Northwest corner:
  0   1   3   4   6   8    9    11   12   14   16
  1   2   4   5   7   9    10   12   13   15   17
  2   3   5   6   8   10   11   13   14   16   18
  3   4   6   7   9   11   12   14   15   17   19
  4   5   7   8   10  12   13   15   16   18   20
  5   6   8   9   11  13   14   16   17   19   21
As a triangle (antidiagonals):
  0
  1   1
  2   2   3
  3   3   4   4
  4   4   5   5   6
  5   5   6   6   7   8
  6   6   7   7   8   9   9

Cf. A000210, A001950, A001622, A084531, A167267, A019446.

t[n_, k_] := Floor[n + k*GoldenRatio];
Grid[Table[t[n, k], {n, 0, 10}, {k, 0, 10}]] (* array *)
u = Table[t[n - k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten  (* sequence *)

T(n,k) = floor(n + k*r), where r = golden ratio = (1+sqrt(5))/2.

A178129 Partial sums of A050508.

Original entry on oeis.org

0, 2, 8, 23, 47, 87, 147, 224, 328, 463, 623, 821, 1049, 1322, 1644, 2004, 2420, 2896, 3418, 4007, 4647, 5361, 6153, 7004, 7940, 8940, 10032, 11220, 12480, 13843, 15313, 16863, 18527, 20276, 22146, 24141, 26229, 28449, 30767, 33224, 35824, 38530

Offset: 0

Jonathan Vos Post, May 20 2010

Partial sums of golden rectangle numbers. The subsequence

			a(19) = 0 + 2 + 6 + 15 + 24 + 40 + 60 + 77 + 104 + 135 +

Cf. A000201, A007064, A026355, A007067, A050508.

from math import isqrt
from itertools import count, islice
def A178129_gen(): # generator of terms
    return accumulate(n*((isqrt(5*n**2<<2)>>1)+n+1>>1) for n in count(0))
A178129_list = list(islice(A178129_gen(),10)) # Chai Wah Wu, Aug 29 2022

a(n) = Sum_{i=0..n} A050508(i) = Sum_{i=0..n} (i*A007067(i)).

cp's OEIS Frontend

A007067 Nearest integer to n*tau where tau = (1+sqrt(5))/2.

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A007066 a(n) = 1 + ceiling((n-1)*phi^2), phi = (1+sqrt(5))/2.

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A099267 Numbers generated by the golden sieve.

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A026356 a(n) = floor((n-1)*phi) + n + 1, n > 0, where phi = (1+sqrt(5))/2.

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A189662 Positions of 0 in A189661; complement of A026356.

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A054082 Permutation of N: a(1)=2, a(2)=1 and for each k >= 2, let p(k)=least natural number not already an a(i), q(k)=p(k)+k-1, a(2k-1)=q(k), a(2k)=p(k).

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A332502 Rectangular array read by antidiagonals: T(n,k) = floor(n + k*r), where r = golden ratio = (1+sqrt(5))/2.

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A178129 Partial sums of A050508.