A105424 -id:A105424 - cp's OEIS frontend

A105425 Interpret A105424 as a binary sequence and convert to decimal.

0, 1, 2, 4, 5, 8, 10, 16, 17, 18, 20, 21, 32, 34, 36, 37, 40, 42, 64, 65, 66, 68, 69, 72, 74, 80, 81, 82, 84, 85, 128, 130, 132, 133, 136, 138, 144, 145, 146, 148, 149, 160, 162, 164, 165, 168, 170, 256, 257, 258, 260, 261, 264, 266, 272, 273, 274

Offset: 0

Bryan Jacobs (bryanjj(AT)gmail.com), Apr 08 2005

See A105424 for further information. - N. J. A. Sloane, Mar 01 2021

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, PARI program for A105425

Cf. A105424.

A105425[n_]:=FromDigits[First[RealDigits[n,GoldenRatio,Floor[Log[GoldenRatio,Max[n,1]]]+1]],2];Array[A105425,100,0] (* Paolo Xausa, Oct 20 2023 *)

PARI
```
See Links section.
```

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

A130600 Integers written in base phi, with the "decimal point" omitted.

Original entry on oeis.org

1, 1001, 10001, 10101, 10001001, 10100001, 100000001, 100010001, 100100101, 101000101, 101010101, 100000101001, 100010001001, 100100001001, 100101001001, 101000100001, 101010000001, 1000000000001, 1000001000001

Offset: 1

Casey Mongoven, Aug 06 2007

This is the "greedy" or "minimal" representation (see also A130601).

			If the decimal point were included, the sequence would read 1., 10.01, 100.01, 101.01, 1000.1001, 1010.0001, 10000.0001, 10001.0001, 10010.0101, 10100.0101, 10101.0101, ... Unfortunately these are not integers.
Examples: a(2)=1001 because phi^1+phi^-2 = 2, a(3) = 10001 because phi^2+phi^-2 = 3, a(4) = 10101 because phi^2+phi^0+phi^-2 = 4.

Casey Mongoven and T. D. Noe, Table of n, a(n) for n = 1..1000
Casey Mongoven, Music based on this sequence
Ron Knott, Integers written in base phi

Cf. A001622, A055778, A105424, A130601.

nn = 100; len = 2*Ceiling[Log[GoldenRatio, nn]]; Table[d = RealDigits[n, GoldenRatio, len]; last1 = Position[d[[1]], 1][[-1, 1]]; FromDigits[Take[d[[1]], last1]], {n, nn}] (* T. D. Noe, May 20 2011 *)

A341722 The part of n in base phi right of the decimal point (reversed), using a greedy algorithm representation (more precisely, using the Bergman-canonical representation).

Original entry on oeis.org

0, 0, 10, 10, 10, 1001, 1000, 1000, 1000, 1010, 1010, 1010, 100101, 100100, 100100, 100100, 100001, 100000, 100000, 100000, 100010, 100010, 100010, 101001, 101000, 101000, 101000, 101010, 101010, 101010, 10010101, 10010100, 10010100, 10010100, 10010001, 10010000, 10010000

Offset: 0

N. J. A. Sloane, Mar 01 2021

A105424 and A105425 give the part of n in base phi left of the decimal point.

			The first few numbers written in base phi are:
0 = 0.
1 = 1.
2 = 10.01
3 = 100.01
4 = 101.01
5 = 1000.1001
6 = 1010.0001
7 = 10000.0001
8 = 10001.0001
9 = 10010.0101
10 = 10100.0101
11 = 10101.0101
12 = 100000.101001
13 = 100010.001001
14 = 100100.001001
15 = 100101.001001
16 = 101000.100001
17 = 101010.000001
18 = 1000000.000001
19 = 1000001.000001
20 = 1000010.010001
21 = 1000100.010001
22 = 1000101.010001
23 = 1001000.100101
24 = 1001010.000101
...

Hugo Pfoertner, Table of n, a(n) for n = 0..1000
F. Michel Dekking, How to add two natural numbers in base phi, arXiv:2002.01665 [math.NT], 5 Feb 2020.
C. Frougny and J. Sakarovitch, Automatic conversion from Fibonacci representation to representation in base phi, and a generalization, Int. J. Algebra Comput. 9 (1999), 351-384. See also preprint.
Ron Knott, Phigits and the Base Phi representation.
Ron Knott, Phigits and the Base Phi representation [Local copy, pdf only]
Jeffrey Shallit, Proving Properties of phi-Representations with the Walnut Theorem-Prover, arXiv:2305.02672 [math.NT], 2023.

Cf. A105424, A105425.

Definition clarified by N. J. A. Sloane, May 27 2023

A104605 Triangle read by rows: row n gives list of powers of phi in the representation of the integer n as a sum of increasing nonconsecutive powers of the golden ratio.

Original entry on oeis.org

0, -2, 1, -2, 2, -2, 0, 2, -4, -1, 3, -4, 1, 3, -4, 4, -4, 0, 4, -4, -2, 1, 4, -4, -2, 2, 4, -4, -2, 0, 2, 4, -6, -3, -1, 5, -6, -3, 1, 5, -6, -3, 2, 5, -6, -3, 0, 2, 5, -6, -1, 3, 5, -6, 1, 3, 5, -6, 6, -6, 0, 6, -6, -2, 1, 6, -6, -2, 2, 6, -6, -2, 0, 2, 6, -6, -4, -1, 3, 6, -6, -4, 1, 3, 6, -6, -4, 4, 6, -6, -4, 0, 4, 6, -6, -4, -2, 1, 4, 6, -6, -4

Offset: 1

Eric W. Weisstein, Mar 17 2005

Let f(n) = F(n+1) = A000045(n) and extend n to include negative indices. Then each row n can equally well be thought of as a sequence a_1, a_2,..., a_k such that f(a_1) + f(a_2) + ... + f(a_k) = n. For example, the fifth row is -4 -1 3, so f(-4) + f(-1) + f(3) = 2 + 0 + 3 = 5. - Dale Gerdemann, Apr 01 2012

			   0
  -2  1
  -2  2
  -2  0  2
  -4 -1  3
  -4  1  3
  -4  4
  -4  0  4
  ...
phi^0, phi^(-2) + phi, phi^(-2) + phi^2, phi^(-2) + phi^0 + phi^2, ...

T. D. Noe, Rows n = 1..1000, flattened
Dale Gerdemann, Combinatorial proofs of Zeckendorf family identities, Fib. Quart. 46/47 (2009) 249.
Eric Weisstein's World of Mathematics, Phi Number System

Cf. A055778 (length of row n), A105424, A178482 (phi-antipalindromic numbers).

nn = 100; len = 2*Ceiling[Log[GoldenRatio, nn]]; Table[d = RealDigits[n, GoldenRatio, len]; Reverse[d[[2]] - Flatten[Position[d[[1]], 1]]], {n, nn}] (* T. D. Noe, May 20 2011 *)

A130601 Integers written in base phi, with the "decimal point" omitted.

Original entry on oeis.org

1, 111, 1101, 10101, 1011111, 1110111, 10101101, 10111101, 11011101, 11110101, 101010101, 10101111111, 10111011111, 11010110111, 11011110111, 11101110111, 11111010111, 101010101101, 101011101101, 101101101101

Offset: 1

Casey Mongoven, Aug 06 2007

The map 100 -> 011 is used to eliminate every 100 from the minimal representation (A130600).

Other sequences are possible for representing integers in base-phi with no occurrence of "00" in any terms - see links.

			Examples: a(2)=111 because phi^0+phi^-1+phi^-2 = 2, a(3) = 1101 because phi^1+phi^0+phi^-2 = 3, a(4) = 10101 because phi^2+phi^0+phi^-2 = 4.

Casey Mongoven, Table of n, a(n) for n = 1..55
L. C. Eggan, C. L. Vanden Eynden, "Decimal" expansions to nonintegral bases, Am. Math. Monthly 73 (6) (1966) 576-582
Ron Knott, Integers written in base phi
Casey Mongoven, Music based on this sequence

Cf. A001622, A055778, A105424, A130600.

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

A105116 The part of n left of the decimal point when written in base e using a greedy algorithm representation.

Original entry on oeis.org

0, 1, 2, 10, 11, 12, 20, 21, 100, 101, 102, 110, 111, 120, 121, 200, 201, 202, 210, 211, 212, 1000, 1001, 1010, 1011, 1012, 1020, 1021, 1100, 1101, 1102, 1110, 1111, 1120, 1121, 1200, 1201, 1202, 1210, 1211, 1212, 2000, 2001, 2010, 2011, 2012, 2020

Offset: 0

Bryan Jacobs (bryanjj(AT)gmail.com), Apr 08 2005

			3 in base e = 10.020... so a(3) = 10.

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A001113 (e digits).
Cf. A105424 (base phi), A344939 (base Pi).
Cf. A363832 (number of digits).

A105116[n_]:=FromDigits[First[RealDigits[n,E,Floor[Log[E,Max[n,1]]]+1]]];
Array[A105116,100,0] (* Paolo Xausa, Oct 18 2023 *)

A344939 The part of n left of the radix point when written in base Pi using a greedy algorithm representation.

Original entry on oeis.org

0, 1, 2, 3, 10, 11, 12, 20, 21, 22, 100, 101, 102, 103, 110, 111, 112, 120, 121, 122, 200, 201, 202, 210, 211, 212, 213, 220, 221, 222, 300, 301, 1000, 1001, 1002, 1010, 1011, 1012, 1020, 1021, 1022, 1100, 1101, 1102, 1103, 1110, 1111, 1112, 1120, 1121, 1122

Offset: 0

Paolo Xausa, Jun 03 2021

			a(5) = 11 because 5 in base Pi is 11.22012202...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Wikipedia, Non-integer base of numeration.

Cf. A000796 (Pi digits). Subsequence of A007090.
Cf. A105116 (base e), A105424 (base phi).
Cf. A366721 (number of digits).

A344939 := proc(n)
    local e,ntrunc,a,d;
    Digits := 1000 ;
    if n = 0 then
        return 0 ;
    end if;
    ntrunc := n ;
    e := floor(log(n)/log(Pi)) ;
    a := 0 ;
    while e >= 0 do
        d := floor(ntrunc/Pi^e) ;
        a := 10*a+d ;
        ntrunc := evalf(ntrunc-d*Pi^e) ;
        e := e-1 ;
    end do:
    a ;
end proc:
seq(A344939(n),n=0..15) ; # R. J. Mathar, Aug 16 2021

A344939[n_]:=FromDigits[First[RealDigits[n,Pi,Floor[Log[Pi,Max[n,1]]]+1]]];
Array[A344939,100,0] (* Paolo Xausa, Oct 17 2023 *)

Name edited by Paolo Xausa, Oct 18 2023

A362917 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

0, 1, 10, 11, 101, 1000, 1010, 1011, 10001, 10010, 10011, 10101, 100000, 100010, 100011, 100101, 101000, 101010, 101011, 1000001, 1000010, 1000011, 1000101, 1001000, 1001010, 1001011, 1010001, 1010010, 1010011, 1010101, 10000000

Offset: 0

N. J. A. Sloane, May 26 2023

The part to the right of the decimal point, reversed, is given by A341722, that is, it is the same as in the Bergman-canonical representation. I asked Jeffrey Shallit to confirm this, and he provided the following verification using the Walnut Theorem-Prover:
[Walnut]$ eval sloane "?msd_fib An,x1,x2,y1,y2 ($saka(n,x1,y1) & $dvl(n,x2,y2)) => $equal(y1,y2)":
(saka(n,x1,y1))&dvl(n,x2,y2))):54 states - 66ms
 ((saka(n,x1,y1))&dvl(n,x2,y2)))=>equal(y1,y2))):2 states - 25ms
  (A n , x1 , x2 , y1 , y2 ((saka(n,x1,y1))&dvl(n,x2,y2)))=>equal(y1,y2)))):1 states - 81ms
Total computation time: 264ms.
__
TRUE

			The canonical base phi representations of the numbers 0 through 12 are:
0 = 0.
1 = 1.
2 = 10.01
3 = 11.01
4 = 101.01
5 = 1000.1001
6 = 1010.0001
7 = 1011.0001
8 = 10001.0001
9 = 10010.0101
10 = 10011.0101
11 = 10101.0101
12 = 100000.101001

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.
Jeffrey Shallit, Proving Properties of phi-Representations with the Walnut Theorem-Prover, arXiv:2305.02672 [math.NT], 2023. (see chapter 10, page 28)

Cf. A341722, A362918.

Differs from A105424 at positions given by A003231.

a(13)-a(32) from Hugo Pfoertner, May 26 2023

cp's OEIS Frontend

A105425 Interpret A105424 as a binary sequence and convert to decimal.

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

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A130600 Integers written in base phi, with the "decimal point" omitted.

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A341722 The part of n in base phi right of the decimal point (reversed), using a greedy algorithm representation (more precisely, using the Bergman-canonical representation).

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A104605 Triangle read by rows: row n gives list of powers of phi in the representation of the integer n as a sum of increasing nonconsecutive powers of the golden ratio.

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A130601 Integers written in base phi, with the "decimal point" omitted.

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

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A105116 The part of n left of the decimal point when written in base e using a greedy algorithm representation.

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