A362917 -id:A362917 - cp's OEIS frontend

A362921 The Dekking-van-Loon-canonical base-phi representation of n described in A362917 written as a binary string, omitting the dot.

0, 1, 1001, 1101, 10101, 10001001, 10100001, 10110001, 100010001, 100100101, 100110101, 101010101, 100000101001, 100010001001, 100011001001, 100101001001, 101000100001, 101010000001, 101011000001, 1000001000001, 1000010010001, 1000011010001, 1000101010001, 1001000100101, 1001010000101

Offset: 0

N. J. A. Sloane, May 27 2023

Michel Dekking and Ad van Loon, On the representation of the natural numbers by powers of the golden mean, arXiv:2111.07544 [math.NT], 15 Nov 2021.
Dekking, Michel, and Ad van Loon. On the representation of the natural numbers by powers of the golden mean, Fib. Quart. 61:2 (May 2023), 105-118.
Jeffrey Shallit, Proving Properties of phi-Representations with the Walnut Theorem-Prover, arXiv:2305.02672 [math.NT], 2023.

Cf. A362917, A341722.

A003231 a(n) = floor(n*(sqrt(5)+5)/2).

Original entry on oeis.org

3, 7, 10, 14, 18, 21, 25, 28, 32, 36, 39, 43, 47, 50, 54, 57, 61, 65, 68, 72, 75, 79, 83, 86, 90, 94, 97, 101, 104, 108, 112, 115, 119, 123, 126, 130, 133, 137, 141, 144, 148, 151, 155, 159, 162, 166, 170, 173, 177, 180, 184, 188, 191, 195, 198, 202, 206, 209

Offset: 1

N. J. A. Sloane

Let r = (5 - sqrt(5))/2 and s = (5 + sqrt(5))/2. Then 1/r + 1/s = 1, so that A249115 and A003231 are a pair of complementary Beatty sequences. Let tau = (1 + sqrt(5))/2, the golden ratio. Let R = {h*tau, h >= 1} and S = {k*(tau - 1), k >= 1}. Then A003231(n) is the position of n*tau in the ordered union of R and S. The position of n*(tau - 1) is A249115(n). - Clark Kimberling, Oct 21 2014
This is the function named c in the Carlitz-Scoville-Vaughan link. - Eric M. Schmidt, Aug 06 2015
Natural numbers whose representation in base phi differs between the Bergmann representation and the "canonical" representation described by Dekking and van Loon. See proposition 3.3 in Dekking, van Loon (2021). - Hugo Pfoertner, May 26 2023

Dekking, Michel, and Ad van Loon. "On the representation of the natural numbers by powers of the golden mean." arXiv preprint arXiv:2111.07544 (2021); Fib. Quart. 61:2 (May 2023), 105-118.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

L. Carlitz, R. Scoville and T. Vaughan, Some arithmetic functions related to Fibonacci numbers, Fib. Quart., 11 (1973), 337-386.
Michel Dekking and Ad van Loon, On the representation of the natural numbers by powers of the golden mean, arXiv:2111.07544 [math.NT], 15 Nov 2021.
Scott V. Tezlaf, On ordinal dynamics and the multiplicity of transfinite cardinality, arXiv:1806.00331 [math.NT], 2018. See p. 9.
Index entries for sequences related to Beatty sequences

Cf. A000201, A003231, A003233, A003234, A249115.

Cf. A105424, A362917.

a003231 = floor . (/ 2) . (* (sqrt 5 + 5)) . fromIntegral
-- Reinhard Zumkeller, Oct 03 2014

[Floor(n*(Sqrt(5)+5)/2): n  in [1..100]]; // Vincenzo Librandi, Oct 23 2014

A003231:=n->floor(n*(sqrt(5)+5)/2): seq(A003231(n), n=1..100); # Wesley Ivan Hurt, Aug 06 2015

With[{c=(Sqrt[5]+5)/2}, Floor[c*Range[60]]] (* Harvey P. Dale, Oct 01 2012 *)

PARI
```
a(n)=floor(n*(sqrt(5)+5)/2)
```

a(n)=(5*n+sqrtint(5*n^2))\2; \\ Michel Marcus, Nov 14 2023

from math import isqrt
def A003231(n): return (n+isqrt(5*n**2)>>1)+(n<<1) # Chai Wah Wu, Aug 25 2022

a(n) = 2*n + A000201(n). - R. J. Mathar, Aug 22 2014

Better description and more terms from Michael Somos, Jun 07 2000

A362692 Length of the "integer part" of the phi-expansion of n.

Original entry on oeis.org

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

Offset: 0

Jeffrey Shallit, May 01 2023

nonn

The phi-representation of n is the (essentially) unique way to write n = Sum_{j=L..R} b(j)*phi^j, where b(j) is in {0,1} and -oo < L <= 0 <= R, where phi = (1+sqrt(5))/2, subject to the condition that b(j)b(j+1) != 1. The "integer" part is the string of bits b(R)b(R-1)...b(1)b(0), and its length is thus R+1.

The gaps between consecutive terms are all either 0 or 1, and a gap of 1 occurs if and only if n = 1 or n = L(2i) or n = L(2i-1) + 1 for i >= 1. This is equivalent to Theorem 2.1 of Sanchis and Sanchis (2001).

			For n = 20 we have n = phi^6 + phi^1 + phi^(-2) + phi^(-6), and the "integer part" has largest term phi^6, so a(20) = 7.

Paolo Xausa, Table of n, a(n) for n = 0..10000
George Bergman, A number system with an irrational base, Math. Mag. 31 (1957), 98-110.
G. R. Sanchis and L. A. Sanchis, On the frequency of occurrence of α^i in the α-expansions of the positive integers, Fibonacci Quart. 39 (2001), 123-137.
Jeffrey Shallit, Proving Properties of phi-Representations with the Walnut Theorem-Prover, arXiv:2305.02672 [math.NT], 2023.

Cf. A001622, A105424, A362716, A341722, A362872, A362917.

A362692[n_]:=Floor[Log[GoldenRatio,Max[n,1]]]+1;Array[A362692,100,0] (* Paolo Xausa, Oct 19 2023 *)

There is a linear representation of rank 9 for a(n).

a(n) = ceiling(log_phi(n)) for n >= 2.

a(0) changed to 1 by N. J. A. Sloane, May 26 2023

A362918 Length of the part of n to the left of the decimal point in the Dekking-van-Loon-canonical base phi representation of n.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 4, 4, 5, 5, 5, 5, 6

Offset: 0

N. J. A. Sloane, May 26 2023

a(n) = length of the binary string A362917(n).

Dekking, Michel, and Ad van Loon. "On the representation of the natural numbers by powers of the golden mean." arXiv preprint arXiv:2111.07544 (2021); Fib. Quart. 61:2 (May 2023), 105-118.

Michel Dekking and Ad van Loon, On the representation of the natural numbers by powers of the golden mean, arXiv:2111.07544 [math.NT], 15 Nov 2021.

Cf. A105424, A341722, A362917.

cp's OEIS Frontend

A362921 The Dekking-van-Loon-canonical base-phi representation of n described in A362917 written as a binary string, omitting the dot.

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A003231 a(n) = floor(n*(sqrt(5)+5)/2).

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A362692 Length of the "integer part" of the phi-expansion of n.

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Mathematica

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A362918 Length of the part of n to the left of the decimal point in the Dekking-van-Loon-canonical base phi representation of n.

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