A104605 -id:A104605 - cp's OEIS frontend

A105424 The part of n in base phi left of the decimal point, using a greedy algorithm representation (more precisely, using the Bergman-canonical representation).

0, 1, 10, 100, 101, 1000, 1010, 10000, 10001, 10010, 10100, 10101, 100000, 100010, 100100, 100101, 101000, 101010, 1000000, 1000001, 1000010, 1000100, 1000101, 1001000, 1001010, 1010000, 1010001, 1010010, 1010100, 1010101, 10000000

Offset: 0

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

			2 = 10.01 in base phi, so left of the decimal point is 10.
The first few numbers written in base phi:
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
...

T. D. Noe, 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.
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. [Note that this document has been revised multiple times.]

Cf. A001622, A055778, A105425, A104605.
See A341722 for the part to the right of the decimal point.
Cf. A105116 (base e), A344939 (base Pi).

nn = 1000; len = 2*Ceiling[Log[GoldenRatio, nn]]; Table[d = RealDigits[n, GoldenRatio, len]; FromDigits[Take[d[[1]], d[[2]]]], {n, 0, nn}] (* T. D. Noe, May 20 2011 *)

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

A178482 Phi-antipalindromic numbers.

Original entry on oeis.org

1, 3, 4, 7, 8, 10, 11, 18, 19, 21, 22, 25, 26, 28, 29, 47, 48, 50, 51, 54, 55, 57, 58, 65, 66, 68, 69, 72, 73, 75, 76, 123, 124, 126, 127, 130, 131, 133, 134, 141, 142, 144, 145, 148, 149, 151, 152, 170, 171, 173, 174

Offset: 1

Vladimir Shevelev, May 28 2010

We call m a phi-antipalindromic number if for the vector (a,...,b) (a<...=2, either a(n)+1 or a(n)-1 is in the sequence; also either a(n)+3 or a(n)-3 is in the sequence.
Conjecture: this is the sequence of numbers k for which f(k) is an integer, where f(x) is the change-of-base function defined at A214969 using b=phi and c=b^2. - Clark Kimberling, Oct 17 2012
There is a 21-state automaton accepting the Zeckendorf representations of those n in this sequence. - Jeffrey Shallit, May 03 2023
Kimberling's conjecture has been proven by Ingrid Vukusic and myself. Along the way we prove an alternate characterization of the sequence:  they are the positive integers whose base-phi expansion consists only of even exponents of phi. - Jeffrey Shallit, Aug 28 2025
Alternatively, this sequence consists of those numbers k such that either k or k-1 can be written as the (possibly empty) sum of distinct Lucas numbers L_i where i>=2 and i is even. - Jeffrey Shallit, Aug 28 2025

			The vectors of exponents of 4 and 5 are (-2,0,2) and (-4,-1,3) correspondingly (cf.A104605). Therefore by definition 4 is a phi-antipalindromic number, while 5 is not. Let n=38. Then k=5. Thus a(38)=A005248(5)+a(6)=123+10=133. The vector of exponents of phi in the base-phi expansion of 133 is (-10,-4,-2,2,4,10).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..3071 from R. J. Mathar)
Jeffrey Shallit, Proving Properties of phi-Representations with the Walnut Theorem-Prover, arXiv:2305.02672 [math.NT], 2023.

Cf. A005248, A055778, A104605, A104626, A104627, A104628.

For bisections see A171070, A171071.

phiAPQ[1] = True; phiAPQ[n_] := Module[{d = RealDigits[n, GoldenRatio, 2*Ceiling[Log[GoldenRatio, n]]]}, e = d[[2]] - Flatten @ Position[d[[1]], 1]; Reverse[e] == -e]; Select[Range[200], phiAPQ] (* Amiram Eldar, Apr 23 2020 *)

For k>=1, a(2^k)=A005248(k); if 2^k

A118240 The part of n in base phi left of the decimal using a least-greedy algorithm representation.

Original entry on oeis.org

0, 1, 1, 10, 11, 101, 111, 1010, 1011, 1101, 1110, 1111, 10101, 10111, 11010, 11011, 11101, 11111, 101010, 101011, 101101, 101110, 101111, 110101, 110111, 111010, 111011, 111101, 111110, 111111, 1010101, 1010111, 1011010, 1011011, 1011101

Offset: 0

Graeme McRae, Apr 17 2006

Uses least-greedy algorithm (start with largest possible power of phi, writing a 1 only when required, then work downward).

a(n) is also the left portion of the base-phi representation of n in Knott's representation which uses the least number of 0's, the most 1's, and in which the right-hand portion (see A362919) is finite. - N. J. A. Sloane, May 27 2023

			6 = 111.01101010... in base phi using the least-greedy algorithm. The part to the left of the decimal is a(6) = 111.

Ron Knott, Phigits and the Base Phi representation.
Jeffrey Shallit, Proving Properties of phi-Representations with the Walnut Theorem-Prover, arXiv:2305.02672 [math.NT], 2023. [Note that this document has been revised multiple times.]

Cf. A055778, A104605, A118241, A105424, A362919, A362920.

constant (float): phi=(sqrt(5)+1)/2;
variable (float): lphi=phi^floor[log(n)/log(phi)];
variable (float): rem=n;
variable (integer): count=0;
loop: while lphi>1 {count=count*10; lphi=lphi/phi; if(rem > lphi*phi) { rem=rem-lphi; count++;}}

a(1) corrected by N. J. A. Sloane, May 27 2023

A178493 Numbers of powers of phi in base-phi expansion of phi-antipalindromic numbers (A178482).

Original entry on oeis.org

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

Offset: 1

Vladimir Shevelev, May 28 2010, May 30 2010

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Eric Weisstein's World of Mathematics, Phi Number System

Cf. A178482, A005248, A055778, A104605, A104626, A104627, A104628.

powNum[1] = 1; powNum[n_] := Module[{d = RealDigits[n, GoldenRatio, 2*Ceiling[Log[GoldenRatio, n]]]}, e = d[[2]] - Flatten @ Position[d[[1]], 1]; If[Reverse[e] == -e, Length[e], 0]]; Select[Array[powNum, 400], # > 0 &] (* Amiram Eldar, Apr 23 2020 *)

Extended by T. D. Noe, May 20 2011

A362755 Irregular triangle read by rows; the n-th row lists the numbers k such that if phi^e appears in the base phi expansion of k then phi^e also appears in the base phi expansion of n (where phi denotes A001622, the golden ratio).

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 3, 0, 1, 3, 4, 0, 5, 0, 6, 0, 7, 0, 1, 7, 8, 0, 2, 7, 9, 0, 3, 7, 10, 0, 1, 3, 4, 7, 8, 10, 11, 0, 12, 0, 13, 0, 14, 0, 1, 14, 15, 0, 16, 0, 17, 0, 18, 0, 1, 18, 19, 0, 2, 18, 20, 0, 3, 18, 21, 0, 1, 3, 4, 18, 19, 21, 22, 0, 5, 18, 23, 0, 6, 18, 24

Offset: 0

Rémy Sigrist, May 02 2023

See A361755 for a similar sequence.

			Triangle begins:
  n   n-th row
  --  ------------------------
   0  0
   1  0, 1
   2  0, 2
   3  0, 3
   4  0, 1, 3, 4
   5  0, 5
   6  0, 6
   7  0, 7
   8  0, 1, 7, 8
   9  0, 2, 7, 9
  10  0, 3, 7, 10
  11  0, 1, 3, 4, 7, 8, 10, 11
  12  0, 12
  13  0, 13
  14  0, 14
  15  0, 1, 14, 15

Rémy Sigrist, Table of n, a(n) for n = 0..9999 (rows for n = 0..1060 flattened)
Rémy Sigrist, PARI program
Wikipedia, Golden ratio base

Cf. A001622, A104605, A214971, A361755.

PARI
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See Links section.
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T(n, 1) = 0.

T(n, 2) = 1 iff n belongs to A214971.

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A105424 The part of n in base phi left of the decimal point, using a greedy algorithm representation (more precisely, using the Bergman-canonical representation).

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A178482 Phi-antipalindromic numbers.

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A118240 The part of n in base phi left of the decimal using a least-greedy algorithm representation.

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A178493 Numbers of powers of phi in base-phi expansion of phi-antipalindromic numbers (A178482).

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A362755 Irregular triangle read by rows; the n-th row lists the numbers k such that if phi^e appears in the base phi expansion of k then phi^e also appears in the base phi expansion of n (where phi denotes A001622, the golden ratio).

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