A126714 -id:A126714 - cp's OEIS frontend

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

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

A047924 a(n) = B_{A_n+1}+1, where A_n = floor(nphi) = A000201(n), B_n = floor(nphi^2) = A001950(n) and phi is the golden ratio.

Original entry on oeis.org

3, 6, 11, 14, 19, 24, 27, 32, 35, 40, 45, 48, 53, 58, 61, 66, 69, 74, 79, 82, 87, 90, 95, 100, 103, 108, 113, 116, 121, 124, 129, 134, 137, 142, 147, 150, 155, 158, 163, 168, 171, 176, 179, 184, 189, 192, 197, 202, 205, 210, 213, 218, 223, 226, 231, 234, 239

Offset: 0

N. J. A. Sloane

2nd column of array in A038150.

Apart from the first term also the second column of A126714; see also A223025. - Casey Mongoven, Mar 11 2013

Clark Kimberling, Stolarsky interspersions, Ars Combinatoria 39 (1995), 129-138.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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.
Aviezri S. Fraenkel, Recent results and questions in combinatorial game complexities, Theoretical Computer Science, vol. 249, no. 2 (2000), 265-288.
Aviezri S. Fraenkel, Arrays, numeration systems and Frankenstein games, Theoret. Comput. Sci. 282 (2002), 271-284; preprint.
Clark Kimberling, The first column of an interspersion, The Fibonacci Quarterly 32 (1994), 301-315.

Cf. A007066.

A001950 := proc(n)
        local phi;
        phi := (1+sqrt(5))/2 ;
        floor(n*phi^2) ;
end proc:
A000201 := proc(n)
        local phi;
        phi := (1+sqrt(5))/2 ;
        floor(n*phi) ;
end proc:
A047924 := proc(n)
        1+A001950(1+A000201(n)) ;
end proc: # R. J. Mathar, Mar 20 2013

A[n_] := Floor[n*GoldenRatio]; B[n_] := Floor[n*GoldenRatio^2]; a[n_] := B[A[n]+1]+1; Table[a[n], {n, 0, 56}] (* Jean-François Alcover, Feb 11 2014 *)

from mpmath import *
mp.dps=100
import math
def A(n): return int(math.floor(n*phi))
def B(n): return int(math.floor(n*phi**2))
def a(n): return B(A(n) + 1) + 1 # Indranil Ghosh, Apr 25 2017

from math import isqrt
def A047924(n): return ((m:=(n+isqrt(5*n**2)>>1)+1)+isqrt(5*m**2)>>1)+m+1 # Chai Wah Wu, Aug 25 2022

More terms from Naohiro Nomoto, Jun 08 2001

New description from Aviezri S. Fraenkel, Aug 03 2007

A167198 Fractal sequence of the interspersion A083047.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 4, 2, 3, 5, 1, 4, 6, 2, 7, 3, 5, 8, 1, 9, 4, 6, 10, 2, 7, 11, 3, 12, 5, 8, 13, 1, 9, 14, 4, 15, 6, 10, 16, 2, 17, 7, 11, 18, 3, 12, 19, 5, 20, 8, 13, 21, 1, 22, 9, 14, 23, 4, 15, 24, 6, 25, 10, 16, 26, 2, 17, 27, 7, 28, 11, 18, 29, 3, 30, 12, 19, 31, 5, 20, 32, 8, 33, 13

Offset: 1

Clark Kimberling, Oct 30 2009

nonn

As a fractal sequence, if the first occurrence of each term is deleted, the remaining sequence is the original. In general, the interspersion of a fractal sequence is constructed by rows: row r consists of all n, such that a(n)=r; in particular, A083047 is constructed in this way from A167198.

a(n-1) gives the row number which contains n in the dual Wythoff array A126714 (beginning the row count at 1), see also A223025 and A019586. - Casey Mongoven, Mar 11 2013

			To produce row 5, first write row 4: 2,3,1, then place 4 right before 2, and then place 5 right before 1, getting 4,2,3,5,1.

Clark Kimberling, Stolarsky interspersions, Ars Combinatoria 39 (1995), 129-138.

Clark Kimberling, The first column of an interspersion, The Fibonacci Quarterly 32 (1994), 301-315.

Cf. A003603, A083047, A035513, A000045.

Following is a construction that avoids reference to A083047.
Write initial rows:
Row 1: .... 1
Row 2: .... 1
Row 3: .... 2..1
Row 4: .... 2..3..1
For n>=4, to form row n+1, let k be the least positive integer not yet used; write row n, and right before the first number that is also in row n-1, place k; right before the next number that is also in row n-1, place k+1, and continue. A167198 is the concatenation of the rows. (If "before" is replaced by "after", the resulting fractal sequence is A003603, and the associated interspersion is the Wythoff array, A035513.)

A223025 Gives the column number which contains n in the dual Wythoff array (beginning the column count at 1).

Original entry on oeis.org

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

Offset: 1

Casey Mongoven, Mar 11 2013

nonn

			a(23) = 3 because 23 is in the third column of the dual Wythoff array (see A126714).

Clark Kimberling, Stolarsky interspersions, Ars Combinatoria 39 (1995), 129-138.

Clark Kimberling, The first column of an interspersion, The Fibonacci Quarterly 32 (1994), 301-315.

Cf. A007066, A126714, A047924, A167198, A035614.

cp's OEIS Frontend

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

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A047924 a(n) = B_{A_n+1}+1, where A_n = floor(n*phi) = A000201(n), B_n = floor(n*phi^2) = A001950(n) and phi is the golden ratio.

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A167198 Fractal sequence of the interspersion A083047.

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Formula

A223025 Gives the column number which contains n in the dual Wythoff array (beginning the column count at 1).

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A047924 a(n) = B_{A_n+1}+1, where A_n = floor(nphi) = A000201(n), B_n = floor(nphi^2) = A001950(n) and phi is the golden ratio.