A066096 -id:A066096 - cp's OEIS frontend

A004931 a(n) = floor(n*phi^16), where phi is the golden ratio, A001622.

0, 2206, 4413, 6620, 8827, 11034, 13241, 15448, 17655, 19862, 22069, 24276, 26483, 28690, 30897, 33104, 35311, 37518, 39725, 41932, 44139, 46346, 48553, 50760, 52967, 55174, 57381, 59588, 61795

Offset: 0

N. J. A. Sloane

nonn

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A004919, A004920, A004921, A004922, A004923, A004924, A004925.
Cf. A004926, A004927, A004928, A004929, A004930, A004932, A004933.
Cf. A004934, A004935, A004976, A066096, A090909.
Cf. A001622.

[Floor((2207+987*Sqrt(5))*n/2): n in [0..60]]; // G. C. Greubel, Sep 06 2023

Floor[GoldenRatio^(16)*Range[0, 60]] (* G. C. Greubel, Sep 06 2023 *)

[floor(golden_ratio^(16)*n) for n in range(61)] # G. C. Greubel, Sep 06 2023

A004933 a(n) = floor(n*phi^18), where phi is the golden ratio, A001622.

Original entry on oeis.org

0, 5777, 11555, 17333, 23111, 28889, 34667, 40445, 46223, 52001, 57779, 63557, 69335, 75113, 80891, 86669, 92447, 98225, 104003, 109781, 115559, 121337, 127115, 132893, 138671, 144449, 150227

Offset: 0

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A004919, A004920, A004921, A004922, A004923, A004924, A004925.
Cf. A004926, A004927, A004928, A004929, A004930, A004931, A004932.
Cf. A004934, A004935, A004976, A066096, A090909.
Cf. A001622.

[Floor((2889+1292*Sqrt(5))*n): n in [0..60]]; // G. C. Greubel, Sep 11 2023

With[{p=GoldenRatio^18},Floor[p*Range[0,30]]] (* Harvey P. Dale, Mar 06 2022 *)

[floor(golden_ratio^(18)*n) for n in range(61)] # G. C. Greubel, Sep 11 2023

A004935 a(n) = floor(n*phi^20), where phi is the golden ratio, A001622.

Original entry on oeis.org

0, 15126, 30253, 45380, 60507, 75634, 90761, 105888, 121015, 136142, 151269, 166396, 181523, 196650, 211777, 226904, 242031, 257158, 272285, 287412, 302539, 317666, 332793, 347920, 363047, 378174

Offset: 0

N. J. A. Sloane

From Joerg Arndt, Sep 12 2023: (Start)
phi^20 = 15126.999933893... is a near integer.
Therefore the (incorrect!) g.f. 1 + (-1 + 15128*x)/(1-x)^2 produces the initial about 15000 terms of this sequence.
(End)

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for sequences related to Beatty sequences

Cf. A004919, A004920, A004921, A004922, A004923, A004924, A004925.
Cf. A004926, A004927, A004928, A004929, A004930, A004931, A004932.
Cf. A004933, A004934, A004976, A066096, A090909.
Cf. A001622.

[Floor((15127+6765*Sqrt(5))*n/2): n in [0..60]]; // G. C. Greubel, Sep 12 2023

With[{c=GoldenRatio^20},Floor[c Range[0,30]]] (* Harvey P. Dale, Feb 18 2013 *)

[floor(golden_ratio^(20)*n) for n in range(61)] # G. C. Greubel, Sep 12 2023

A060143 a(n) = floor(n/tau), where tau = (1 + sqrt(5))/2.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 9, 10, 11, 11, 12, 12, 13, 14, 14, 15, 16, 16, 17, 17, 18, 19, 19, 20, 21, 21, 22, 22, 23, 24, 24, 25, 25, 26, 27, 27, 28, 29, 29, 30, 30, 31, 32, 32, 33, 33, 34, 35, 35, 36, 37, 37, 38, 38, 39, 40, 40, 41, 42, 42, 43, 43, 44, 45, 45

Offset: 0

Clark Kimberling, Mar 05 2001

Fibonacci base shift right: for n >= 0, a(n+1) = Sum_{k in A_n} F_{k-1}, where n = Sum_{k in A_n} F_k (unique) expression of n as a sum of "noncontiguous" Fibonacci numbers (with index >=2). - Michele Dondi (bik.mido(AT)tiscalenet.it), Dec 30 2001 [corrected, and aligned with sequence offset by Peter Munn, Jan 10 2018]
Numerators a(n) of fractions slowly converging to phi, the golden ratio: let a(1) = 0, b(n) = n - a(n); if (a(n) + 1) / b(n) < (1 + sqrt(5))/2, then a(n+1) = a(n) + 1, else a(n+1) = a(n). a(n) + b(n) = n and as n -> +infinity, a(n) / b(n) converges to (1 + sqrt(5))/2. For all n, a(n) / b(n) < (1 + sqrt(5))/2. - Robert A. Stump (bee_ess107(AT)msn.com), Sep 22 2002
a(10^n) gives the first few digits of phi=(sqrt(5)-1)/2.
Comment corrected, two alternative ways, by Peter Munn, Jan 10 2018: (Start)
(a(n) = a(n+1) or a(n) = a(n-1)) if and only if a(n) is in A066096.
a(n+1) = a(n+2) if and only if n is in A003622.
(End)
From Wolfdieter Lang, Jun 28 2011: (Start)
a(n+1) counts for n >= 1 the number of Wythoff A-numbers not exceeding n.
a(n+1) counts also the number of Wythoff B-numbers smaller than A(n+2), with the Wythoff A- and B-sequences A000201 and A001950, respectively.
a(n+1) = Sum_{j=1..n} A005614(j-1) for n >= 1 (no rounding problems like in the above definition, because the rabbit sequence A005614(n-1) for n >= 1, can be defined by a substitution rule).
a(n+1) = A(n+1)-(n+1) (serving, together with the last equation, as definition for A(n+1), given the rabbit sequence).
a(n+1) = A005206(n), n >= 0.
(End)
Let b(n) = floor((n+1)/phi). Then b(n) + b(b(n-1)) = n [Granville and Rasson]. - N. J. A. Sloane, Jun 13 2014

			a(6)= 3 so b(6) = 6 - 3 = 3. a(7) = 4 because (a(6) + 1) / b(6) = 4/3 which is < (1 + sqrt(5))/2. So b(7) = 7 - 4 = 3. a(8) = 4 because (a(7) + 1) / b(7) = 5/3 which is > (1 + sqrt(5))/2. - Robert A. Stump (bee_ess107(AT)msn.com), Sep 22 2002
From _Wolfdieter Lang_, Jun 28 2011: (Start)
There are a(4) = 2 (positive) Wythoff A-numbers <= 3, namely 1 and 3.
There are a(4) = 2 (positive) Wythoff B-numbers < A(4) = 6, namely 2 and 5.
a(4) = 2 = A(4) - 4 = 6 - 4.
(End)

William A. Tedeschi, Table of n, a(n) for n = 0..10000 [This replaces an earlier b-file computed by Harry J. Smith]
M. Celaya and F. Ruskey (Proposers), Another property of only the golden ratio, Problem 11651, Amer. Math. Monthly, 121 (2014), 550-551.
F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
Vincent Granville and Jean-Paul Rasson, A strange recursive relation, J. Number Theory 30 (1988), no. 2, 238--241. MR0961919(89j:11014). - From _N. J. A. Sloane_, Jun 13 2014
Jeffrey Shallit, The Hurt-Sada Array and Zeckendorf Representations, arXiv:2501.08823 [math.NT], 2025. See p. 2.

Cf. A000045 (Fibonacci numbers), A003622, A022342, A035336.
Terms that occur only once: A001950.
Terms that occur twice: A066096 (a version of A000201).
Numerator sequences for other values, as described in Robert A. Stump's 2002 comment: A074065 (sqrt(3)), A074840 (sqrt(2)).
Apart from initial terms, same as A005206.
First differences: A096270 (a version of A005614).
Partial sums: A183136.

[Floor(2*n/(1+Sqrt(5))): n in [0..80]]; // Vincenzo Librandi, Mar 29 2015

Floor[Range[0,80]/GoldenRatio] (* Harvey P. Dale, May 09 2013 *)

{ default(realprecision, 10); p=(sqrt(5) - 1)/2; for (n=0, 1000, write("b060143.txt", n, " ", floor(n*p)); ) } \\ Harry J. Smith, Jul 02 2009

from math import isqrt
def A060143(n): return (n+isqrt(5*n**2)>>1)-n # Chai Wah Wu, Aug 10 2022

a(n) = floor(phi(n)), where phi=(sqrt(5)-1)/2. [corrected by Casey Mongoven, Jul 18 2008]
a(F_n + 1) = F_{n-1} if F_n is the n-th Fibonacci number. [aligned with sequence offset by Peter Munn, Jan 10 2018]
a(1) = 0. b(n) = n - a(n). If (a(n) + 1) / b(n) < (1 + sqrt(5))/2, then a(n+1) = a(n) + 1, else a(n+1) = a(n). - Robert A. Stump (bee_ess107(AT)msn.com), Sep 22 2002 [corrected by Peter Munn, Jan 07 2018]
A006336(n) = A006336(n-1) + A006336(a(n)) for n>1. - Reinhard Zumkeller, Oct 24 2007
a(n) = floor(n*phi) - n, where phi = (1+sqrt(5))/2. - William A. Tedeschi, Mar 06 2008
Celaya and Ruskey give an interesting formula for a(n). - N. J. A. Sloane, Jun 13 2014

I merged three identical sequences to create this entry. Some of the formulas may need their initial terms adjusting now. - N. J. A. Sloane, Mar 05 2003

More terms from William A. Tedeschi, Mar 06 2008

A082389 a(n) = floor((n+2)phi) - floor((n+1)phi) where phi=(1+sqrt(5))/2.

Original entry on oeis.org

1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2

Offset: 1

Benoit Cloitre, Apr 14 2003

Alternative descriptions (1): unique positive integer sequence taking values in {1,2} satisfying a(1)=1, a(2)=2 and a(a(1)+...+a(n))=a(n) for n >= 3.
(2) Start with 1,2; then for any k>=1, a(a(1)+...+a(k))=a(k), fill in any undefined terms by the rule that a(t) = 1 if a(t-1) = 2 and a(t) = 2 if a(t-1) = 1.
(3) a(1)= 1, a(2)=2, a(a(1)+a(2)+...+a(n))=a(n); a(a(1)+a(2)+...+a(n)+1)=3-a(n).
More generally, the sequence a(n)=floor(r*(n+2))-floor(r*(n+1)), r= (1/2) *(z+sqrt(z^2+4)), z integer >=1, is defined by a(1), a(2) and a(a(1)+a(2)+...+a(n)+f(z))=a(n); a(a(1)+a(2)+...+a(n)+f(z)+1)=(2z+1)-a(n) where f(1)=0, f(z)=z-2 for z>=2.

			a(1)+a(2)=3 and a(a(1)+a(2)) must be a(2) so a(3)=2. Therefore a(a(1)+a(2)+a(3))=a(5)=2 and from the rule the "hole" a(4) is 1. Hence sequence begins 1,2,2,1,2,...

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Same as A014675 without the first term.

A082389:=n->floor((n+2)*(1+sqrt(5))/2) - floor((n+1)*(1+sqrt(5))/2): seq(A082389(n), n=1..300); # Wesley Ivan Hurt, Jan 16 2017

Rest@Nest[ Flatten[ # /. {1 -> 2, 2 -> {2, 1}}] &, {1}, 11] (* Robert G. Wilson v, Jan 26 2006 *)
#[[2]]-#[[1]]&/@Partition[Table[Floor[GoldenRatio n],{n,0,110}],2,1] (* Harvey P. Dale, Sep 04 2019 *)
Differences[Floor[GoldenRatio Range[2,150]]] (* Harvey P. Dale, Dec 02 2024 *)

from math import isqrt
def A082389(n): return (n+2+isqrt(m:=5*(n+2)**2)>>1)-(n+1+isqrt(m-10*n-15)>>1) # Chai Wah Wu, Aug 29 2022

a(n) = A014675(n+1); sum(k = 1, n, a(k)) = A058065(n)
Apparently a(n) = A059426(n).
a(n) = A066096(n+2)-A066096(n+1). - R. J. Mathar, Aug 02 2024

A381110 a(n) is the maximum number of points from the set {(k, f(k)); k = 0..n} belonging to a straight line passing through the point (n, f(n)), where f(n) = A060143(n) = floor(n/phi) and phi is the golden ratio (sqrt(5)+1)/2.

Original entry on oeis.org

1, 2, 2, 2, 3, 3, 4, 3, 5, 3, 4, 4, 4, 5, 3, 5, 4, 4, 6, 4, 5, 5, 5, 6, 5, 6, 7, 6, 5, 6, 7, 6, 7, 5, 7, 8, 6, 8, 6, 7, 9, 6, 9, 7, 6, 10, 6, 7, 8, 7, 11, 7, 7, 9, 7, 12, 7, 8, 10, 8, 8, 8, 8, 11, 8, 9, 9, 9, 9, 8, 9, 10, 9, 10, 9, 10, 11, 8, 10, 10, 10, 11, 8

Offset: 0

Pontus von Brömssen, Feb 14 2025

The sequence would remain the same if A060143 in the definition were replaced with A066096, i.e., if points (k, floor(k*phi)) were considered instead of (k, floor(k/phi)).

Pontus von Brömssen, Table of n, a(n) for n = 0..10000

Cf. A060143, A066096, A375422, A381111.

A000202 a(8i+j) = 13i + a(j), where 1<=j<=8.

Original entry on oeis.org

1, 3, 4, 6, 8, 9, 11, 12, 14, 16, 17, 19, 21, 22, 24, 25, 27, 29, 30, 32, 34, 35, 37, 38, 40, 42, 43, 45, 47, 48, 50, 51, 53, 55, 56, 58, 60, 61, 63, 64, 66, 68, 69, 71, 73, 74, 76, 77, 79, 81, 82, 84, 86, 87, 89, 90, 92, 94, 95, 97, 99, 100, 102, 103, 105, 107, 108, 110

Offset: 1

N. J. A. Sloane

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Michael De Vlieger, Table of n, a(n) for n = 1..10000
James F. Peters, Problem H-327, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 19, No. 2 (1981), p. 189; Are You Curious?, Solution to Problem H-327 by Paul S. Bruckman, ibid., Vol. 20, No. 4 (1982), pp. 373-375.
D. E. Thoro, Problem H-12, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 1, No. 2 (1963), p. 54; A Curious Sequence, Solution to Problem H-12 by Malcolm Tallman, ibid., Vol. 1, No. 4 (1963), p. 50.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,1,-1).

Different from A000201, A066096, A090908.

Cf. A000045.

a[0] := 0:a[1] := 1:a[2] := 3:a[3] := 4:a[4] := 6:a[5] := 8:a[6] := 9:a[7] := 11:a[8] := 12: for m from 9 to 200 do if irem(m,8)=0 then myrem := 8; myquo := iquo(m,8)-1; else myrem := irem(m,8); myquo := iquo(m,8) fi; a[m] := 13*myquo +a[myrem] od: for k from 1 to 200 do printf(`%a,`,a[k]) od: # James Sellers, May 29 2000

Set[#, {1, 3, 4, 6, 8, 9, 11, 12}] &@ Map[a[#] &, Range[0, 7]]; a[n_] := a[n] = 13 #1 + a[#2] & @@ QuotientRemainder[n, 8]; Array[a, 68, 0] (* Michael De Vlieger, Sep 08 2017 *)

a(n) = floor((13*n - 1)/8); \\ Jon E. Schoenfield, Aug 21 2022

a(n) = floor((13*n - 1)/8). - Jon E. Schoenfield, Aug 21 2022

a(Fibonacci(n)-1) = Fibonacci(n+1) - 2, for n>=6 (Peters, 1981). - Amiram Eldar, Jan 27 2022

More terms from James Sellers, May 29 2000

cp's OEIS Frontend

A004931 a(n) = floor(n*phi^16), where phi is the golden ratio, A001622.

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A004933 a(n) = floor(n*phi^18), where phi is the golden ratio, A001622.

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A004935 a(n) = floor(n*phi^20), where phi is the golden ratio, A001622.

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A060143 a(n) = floor(n/tau), where tau = (1 + sqrt(5))/2.

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A082389 a(n) = floor((n+2)*phi) - floor((n+1)*phi) where phi=(1+sqrt(5))/2.

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A381110 a(n) is the maximum number of points from the set {(k, f(k)); k = 0..n} belonging to a straight line passing through the point (n, f(n)), where f(n) = A060143(n) = floor(n/phi) and phi is the golden ratio (sqrt(5)+1)/2.

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A000202 a(8i+j) = 13i + a(j), where 1<=j<=8.

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A082389 a(n) = floor((n+2)phi) - floor((n+1)phi) where phi=(1+sqrt(5))/2.