A004932 -id:A004932 - cp's OEIS frontend

A066096 a(n) = floor(n*phi), where phi = (1 + sqrt(5))/2.

0, 1, 3, 4, 6, 8, 9, 11, 12, 14, 16, 17, 19, 21, 22, 24, 25, 27, 29, 30, 32, 33, 35, 37, 38, 40, 42, 43, 45, 46, 48, 50, 51, 53, 55, 56, 58, 59, 61, 63, 64, 66, 67, 69, 71, 72, 74, 76, 77, 79, 80, 82, 84, 85, 87, 88, 90, 92, 93, 95, 97, 98, 100, 101, 103, 105, 106

Offset: 0

Michele Dondi (bik.mido(AT)tiscalenet.it), Dec 30 2001

a(n) is the smallest number different from a(i) and a(i)+i for i < n.

The losing positions in the game of Wythoff-Nim are precisely the pairs (a(n), a(n)+n).

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

Essentially the partial sums of A001468.
Cf. A000201, A001622, A003622, A022342, A026351, A035336, A060143.
Cf. A004919, A004920, A004921, A004922, A004923, A004924, A004925.
Cf. A004926, A004927, A004928, A004929, A004930, A004931, A004932.
Cf. A004933, A004934, A004935, A004976, A090909.

[Floor((1+Sqrt(5))*n/2): n in [0..80]]; // G. C. Greubel, Sep 12 2023

Floor[GoldenRatio*Range[0, 80]] (* G. C. Greubel, Sep 12 2023 *)

a(n) = (n+sqrtint(5*n^2))\2;
[a(n)|n<-[0..100]] \\ Simon Strandgaard, Jun 28 2022

[floor(golden_ratio*n) for n in range(81)] # G. C. Greubel, Sep 12 2023

For n >= 1, a(n) = A000201(n).
Duplicate values in A060143.
a(n) = 1 + A022342(n) = A000201(n).
a(n) = floor(n*phi), where phi = (1 + sqrt(5))/2. - Peter Munn, Jan 12 2018
a(n) = A026351(n) - 1. - Philippe Deléham, Jan 15 2023

Name corrected by Peter Munn, Dec 06 2017

New name using a formula from Peter Munn by Peter Luschny, Jan 18 2023

A004922 a(n) = floor(n*phi^7), where phi is the golden ratio, A001622.

Original entry on oeis.org

0, 29, 58, 87, 116, 145, 174, 203, 232, 261, 290, 319, 348, 377, 406, 435, 464, 493, 522, 551, 580, 609, 638, 667, 696, 725, 754, 783, 812, 841, 871, 900, 929, 958, 987, 1016, 1045, 1074, 1103, 1132, 1161, 1190, 1219, 1248, 1277, 1306, 1335, 1364

Offset: 0

N. J. A. Sloane

nonn

Karl V. Keller, Jr., Table of n, a(n) for n = 0..1000
Index entries for sequences related to Beatty sequences

Cf. A004919, A004920, A004921, A004923, A004924, A004925, A004926.
Cf. A004927, A004928, A004929, A004930, A004931, A004932, A004933.
Cf. A004934, A004935, A004976, A066096, A090909.
Cf. A001622, A004962 (ceiling).

[Floor(n*((1 + Sqrt(5))/2)^7): n in [0..50]]; // Vincenzo Librandi, Jul 22 2015

Table[Floor[n ((1 + Sqrt[5])/2)^7], {n, 0, 50}] (* Vincenzo Librandi, Jul 22 2015 *)

from sympy import sqrt
phi = (1 + sqrt(5))/2
for n in range(0,101): print(int(n*phi**7), end=', ') # Karl V. Keller, Jr., Jul 22 2015

A090909 Terms a(k) of A073869 for which a(k-1) = a(k), and a(k) and a(k+1) are distinct.

Original entry on oeis.org

2, 5, 7, 10, 13, 15, 18, 20, 23, 26, 28, 31, 34, 36, 39, 41, 44, 47, 49, 52, 54, 57, 60, 62, 65, 68, 70, 73, 75, 78, 81, 83, 86, 89, 91, 94, 96, 99, 102, 104, 107, 109, 112, 115, 117, 120, 123, 125, 128, 130, 133, 136, 138, 141, 143, 146, 149, 151, 154, 157, 159, 162

Offset: 1

Amarnath Murthy, Dec 14 2003

nonn

Is this the same as A001950? - Alec Mihailovs (alec(AT)mihailovs.com), Jul 23 2007
Identical to n + A066096(n)? - Ed Russell (times145(AT)hotmail.com), May 09 2009
From Michel Dekking, Dec 18 2024: (Start)
Proof of Mihailovs's conjecture: This follows immediately from the result in my 2023 paper in JIS that A073869 is equal to Hofstadter’s G-sequence A005206, and my recent comment in A005206 on the pairs of duplicate values in A005206.
The answer to Russell’s question is well-known, and also Detlef’s formula is well-known.
Originally, this sequence was given the name “Terms a(k) of A073869 for which a(k-1), a(k) and a(k+1) are distinct.’’ These are the triples (1,2,3),(4,5,6),(6,7,8), (9,10,11), ... occurring at  k = 3, k = 8, k = 11, k = 16,... in A005206. Note that if a duplicate pair (a(m-1), a(m)) is followed directly by another duplicate pair, then a(m-3), a(m-2) and a(m-1) are distinct, and only so. This corresponds to the block 00 occurring in the Fibonacci word obtained by projecting A005206 on the Fibonacci word (see Corollary in my recent comment in A005206). These occurrences are at the Wythoff AB numbers A003623 according to Wolfdieter Lang’s comment in A003623. Conclusion: the sequence of terms a(k) of A073869 for which a(k-1), a(k), and a(k+1) are distinct is given by the Wythoff AB-numbers. (End)

			A073869 = A005206 = 0,1,1,2,3,3,4,4,5,6,6,... The pair (1,1) occurs at k = 2.

G. C. Greubel, Table of n, a(n) for n = 0..10000
F. M. Dekking, On Hofstadter's G-sequence, Journal of Integer Sequences 26 (2023), Article 23.9.2, 1-11.

Cf. A001622, A001950, A002251, A005206, A073869, A090908.
Cf. A004919, A004920, A004921, A004922, A004923, A004924, A004925.
Cf. A004926, A004927, A004928, A004929, A004930, A004931, A004932.
Cf. A004933, A004934, A004935, A004976, A066096.

[Floor((3+Sqrt(5))*n/2): n in [0..80]]; // G. C. Greubel, Sep 12 2023

(* First program *)
A002251= Fold[Append[#1, #2 Ceiling[#2/GoldenRatio] -Total[#1]] &, {1}, Range[2, 500]] - 1; (* Birkas Gyorgy's code of A019444, modified *)
A090909= Join[{0}, Select[Partition[A002251, 2, 1], #[[2]] > #[[1]] &][[All, 2]]] (* G. C. Greubel, Sep 12 2023 *)
(* Second program *)
Floor[GoldenRatio^2*Range[0,80]] (* G. C. Greubel, Sep 12 2023 *)

[floor(golden_ratio^2*n) for n in range(81)] # G. C. Greubel, Sep 12 2023

a(n) = floor(phi^2*n), where phi = (1+sqrt(5))/2. - Gary Detlefs, Mar 10 2011

More terms from R. J. Mathar, Sep 29 2017

Name corrected by Michel Dekking, Dec 13 2024

A004919 a(n) = floor(n*phi^4), where phi is the golden ratio, A001622.

Original entry on oeis.org

0, 6, 13, 20, 27, 34, 41, 47, 54, 61, 68, 75, 82, 89, 95, 102, 109, 116, 123, 130, 137, 143, 150, 157, 164, 171, 178, 185, 191, 198, 205, 212, 219, 226, 233, 239, 246, 253, 260, 267, 274, 281, 287, 294, 301, 308

Offset: 0

N. J. A. Sloane

nonn

The golden section or golden ratio is now usually denoted by "phi", but it in the older literature it was more often denoted by "tau." - N. J. A. Sloane, Feb 17 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
A. J. Hildebrand, Junxian Li, Xiaomin Li, and Yun Xie, Almost Beatty Partitions, arXiv:1809.08690 [math.NT], 2018.
Index entries for sequences related to Beatty sequences

Cf. A004920, A004921, A004922, A004923, A004924, A004925, A004926.
Cf. A004927, A004928, A004929, A004930, A004931, A004932, A004933.
Cf. A004934, A004935, A004976, A066096, A090909.
Cf. A000201, A001950.

[Floor((7+3*Sqrt(5))*n/2): n in [0..60]]; // G. C. Greubel, Aug 22 2023

With[{c=GoldenRatio^4},Floor[c*Range[0,50]]] (* Harvey P. Dale, Apr 11 2012 *)

from math import isqrt
def A004919(n): return (3*n+isqrt(45*n**2)>>1)+(n<<1) # Chai Wah Wu, Aug 17 2022

[floor(golden_ratio^4*n) for n in range(61)] # G. C. Greubel, Aug 22 2023

a(n) = 2*n + floor(3*n*tau). [Formula corrected by Charles R Greathouse IV, Mar 11 2011]

A004924 a(n) = floor(n*phi^9), where phi is the golden ratio, A001622.

Original entry on oeis.org

0, 76, 152, 228, 304, 380, 456, 532, 608, 684, 760, 836, 912, 988, 1064, 1140, 1216, 1292, 1368, 1444, 1520, 1596, 1672, 1748, 1824, 1900, 1976, 2052, 2128, 2204, 2280, 2356, 2432, 2508, 2584, 2660, 2736

Offset: 0

N. J. A. Sloane

nonn

The first differences a(n) - a(n-1) generally equal 76 with exceptions for example at n = 77, 153, 229, 305, 381, 457, ..., 5777, 5854, 5930, .... where they equal 77. - R. J. Mathar, Jan 11 2008

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

Cf. A004919, A004920, A004921, A004922, A004923, A004925, A004926.
Cf. A004927, A004928, A004929, A004930, A004931, A004932, A004933.
Cf. A004934, A004935, A004976, A066096, A090909.
Cf. A001622

[Floor((38+17*Sqrt(5))*n): n in [0..60]]; // G. C. Greubel, Aug 24 2023

Floor[GoldenRatio^9*Range[0, 60]] (* G. C. Greubel, Aug 24 2023 *)

[floor(golden_ratio^9*n) for n in range(61)] # G. C. Greubel, Aug 24 2023

A004926 a(n) = floor(n*phi^11), where phi is the golden ratio, A001622.

Original entry on oeis.org

0, 199, 398, 597, 796, 995, 1194, 1393, 1592, 1791, 1990, 2189, 2388, 2587, 2786, 2985, 3184, 3383, 3582, 3781, 3980, 4179, 4378, 4577, 4776, 4975, 5174, 5373, 5572, 5771, 5970, 6169, 6368, 6567, 6766

Offset: 0

N. J. A. Sloane

nonn

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

Cf. A004919, A004920, A004921, A004922, A004923, A004924, A004925.
Cf. A004927, A004928, A004929, A004930, A004931, A004932, A004933.
Cf. A004934, A004935, A004976, A066096, A090909.
Cf. A001622.

[Floor((199+89*Sqrt(5))*n/2): n in [0..60]]; // G. C. Greubel, Aug 27 2023

Floor[GoldenRatio^(11)*Range[0, 60]] (* G. C. Greubel, Aug 27 2023 *)

[floor(golden_ratio^(11)*n) for n in range(61)] # G. C. Greubel, Aug 27 2023

A004928 a(n) = floor(n*phi^13), where phi is the golden ratio, A001622.

Original entry on oeis.org

0, 521, 1042, 1563, 2084, 2605, 3126, 3647, 4168, 4689, 5210, 5731, 6252, 6773, 7294, 7815, 8336, 8857, 9378, 9899, 10420, 10941, 11462, 11983, 12504, 13025, 13546, 14067, 14588, 15109, 15630, 16151

Offset: 0

N. J. A. Sloane

nonn

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

Cf. A004919, A004920, A004921, A004922, A004923, A004924, A004925.
Cf. A004926, A004927, A004929, A004930, A004931, A004932, A004933.
Cf. A004934, A004935, A004976, A066096, A090909.
Cf. A001622.

[Floor((521+233*Sqrt(5))*n/2): n in [0..60]]; // G. C. Greubel, Sep 05 2023

Floor[GoldenRatio^(13)*Range[0, 60]] (* G. C. Greubel, Sep 05 2023 *)

[floor(golden_ratio^(13)*n) for n in range(61)] # G. C. Greubel, Sep 05 2023

A004930 a(n) = floor(n*phi^15), where phi is the golden ratio, A001622.

Original entry on oeis.org

0, 1364, 2728, 4092, 5456, 6820, 8184, 9548, 10912, 12276, 13640, 15004, 16368, 17732, 19096, 20460, 21824, 23188, 24552, 25916, 27280, 28644, 30008, 31372, 32736, 34100, 35464, 36828, 38192

Offset: 0

N. J. A. Sloane

nonn

G. C. Greubel, Table of n, a(n) for n = 0..10000
Tanya Khovanova, Non Recursions

Cf. A004919, A004920, A004921, A004922, A004923, A004924, A004925.
Cf. A004926, A004927, A004928, A004929, A004931, A004932, A004933.
Cf. A004934, A004935, A004976, A066096, A090909.
Cf. A001622.

[Floor((682+305*Sqrt(5))*n): n in [0..60]]; // G. C. Greubel, Sep 05 2023

Floor[GoldenRatio^(15)*Range[0, 60]] (* G. C. Greubel, Sep 05 2023 *)

[floor(golden_ratio^(15)*n) for n in range(61)] # G. C. Greubel, Sep 05 2023

A004934 a(n) = floor(n*phi^19), where phi is the golden ratio, A001622.

Original entry on oeis.org

0, 9349, 18698, 28047, 37396, 46745, 56094, 65443, 74792, 84141, 93490, 102839, 112188, 121537, 130886, 140235, 149584, 158933, 168282, 177631, 186980, 196329, 205678, 215027, 224376, 233725

Offset: 0

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 0..10000
Tanya Khovanova, Non Recursions

Cf. A004919, A004920, A004921, A004922, A004923, A004924, A004925.
Cf. A004926, A004927, A004928, A004929, A004930, A004931, A004932.
Cf. A004933, A004935, A004976, A066096, A090909.
Cf. A001622.

[Floor((9349+4181*Sqrt(5))*n/2): n in [0..60]]; // G. C. Greubel, Sep 12 2023

Floor[GoldenRatio^(19)*Range[0, 60]] (* G. C. Greubel, Sep 12 2023 *)

[floor(golden_ratio^(19)*n) for n in range(61)] # G. C. Greubel, Sep 12 2023

A004920 a(n) = floor(n*phi^5), where phi is the golden ratio, A001622.

Original entry on oeis.org

0, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 133, 144, 155, 166, 177, 188, 199, 210, 221, 232, 243, 255, 266, 277, 288, 299, 310, 321, 332, 343, 354, 365, 377, 388, 399, 410, 421, 432, 443, 454, 465, 476, 487, 499, 510, 521, 532, 543, 554, 565, 576, 587, 598

Offset: 0

N. J. A. Sloane

nonn

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

Cf. A004919, A004921, A004922, A004923, A004924, A004925, A004926.
Cf. A004927, A004928, A004929, A004930, A004931, A004932, A004933.
Cf. A004934, A004935, A004976, A066096, A090909.
Cf. A001622.

[Floor((11+5*Sqrt(5))*n/2): n in [0..60]]; // G. C. Greubel, Aug 22 2023

Table[Floor[n GoldenRatio^5],{n,0,54}] (* Stefano Spezia, Feb 19 2023 *)

[floor(golden_ratio^5*n) for n in range(61)] # G. C. Greubel, Aug 22 2023

cp's OEIS Frontend

A066096 a(n) = floor(n*phi), where phi = (1 + sqrt(5))/2.

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A004922 a(n) = floor(n*phi^7), where phi is the golden ratio, A001622.

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A090909 Terms a(k) of A073869 for which a(k-1) = a(k), and a(k) and a(k+1) are distinct.

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A004919 a(n) = floor(n*phi^4), where phi is the golden ratio, A001622.

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A004924 a(n) = floor(n*phi^9), where phi is the golden ratio, A001622.

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A004926 a(n) = floor(n*phi^11), where phi is the golden ratio, A001622.

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A004928 a(n) = floor(n*phi^13), where phi is the golden ratio, A001622.

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