

A060143


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


10



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,5


COMMENTS

Fibonacci base shift right: for n >= 0, a(n+1) = Sum_{k in A_n} F_{k1}, 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(n1)) 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 Anumbers not exceeding n.
a(n+1) counts also the number of Wythoff Bnumbers smaller than A(n+2), with the Wythoff A and Bsequences A000201 and A001950, respectively.
a(n+1) = Sum_{j=1..n} A005614(j1) for n >= 1 (no rounding problems like in the above definition, because the rabbit sequence A005614(n1) 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(n1)) = n [Granville and Rasson].  N. J. A. Sloane, Jun 13 2014


LINKS

William A. Tedeschi, Table of n, a(n) for n = 0..10000 [This replaces an earlier bfile 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), 550551.
F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
Vincent Granville, JeanPaul Rasson, A strange recursive relation, J. Number Theory 30 (1988), no. 2, 238241. MR0961919(89j:11014).  From N. J. A. Sloane, Jun 13 2014


FORMULA

a(n) = floor(phi(n)), where phi=(sqrt(5)1)/2. [corrected by Casey Mongoven, Jul 18 2008]
a(F_n + 1) = F_{n1} if F_n is the nth 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(n1) + 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


EXAMPLE

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 Anumbers <= 3, namely 1 and 3.
There are a(4) = 2 (positive) Wythoff Bnumbers < A(4) = 6, namely 2 and 5.
a(4) = 2 = A(4)  4 = 6  4.
(End)


MATHEMATICA

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


PROG

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


CROSSREFS

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.
Sequence in context: A247908 A057363 A073869 * A005206 A057365 A014245
Adjacent sequences: A060140 A060141 A060142 * A060144 A060145 A060146


KEYWORD

nonn,frac,easy


AUTHOR

Clark Kimberling, Mar 05 2001


EXTENSIONS

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


STATUS

approved



