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A060143
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Floor(n/tau), with tau:=(1+sqrt(5))/2.
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8
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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; internal format)
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OFFSET
| 0,5
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COMMENTS
| Fibonacci base shift right: a(n) = Sum_{k in A_n} F_{k-1}, where a(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
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. a(1) = 0. b(n) = n - a(n). If (a(n) + 1) / b(n) < sqrt(3), then a(n+1) = a(n) + 1, else a(n+1) = a(n). - 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.
a(n)=a(n+1) iff n in A066096.
From Wolfdieter Lang, Jun 28 2011: (Start)
a(n+1) counts for n>=1 the number of Wythoff A-numbers not exceeding n.
a(n+1) counts also the number of Wythoff B-numbers smaller than A(n+2), with the Wythoff A- and B-sequences A000201 and A001950, respectively.
a(n+1) = sum(A005614(j-1),j=1..n), n>=1 (no rounding problems like in the above definition, because the rabbit sequence A005614(n-1), 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)
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LINKS
| William A. Tedeschi, Table of n, a(n) for n=0..10000 [This replaces an earlier b-file computed by Harry J. Smith]
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FORMULA
| a(n)=floor(phi(n)), where phi=(sqrt(5)-1)/2. [Corrected by Casey Mongoven (cm(AT)caseymongoven.com), Jul 18 2008]
a(F_n)=F_{n-1} if F_n is the N_th Fibonacci number.
A006336(n) = A006336(n-1) + A006336(a(n)) for n>1. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 24 2007
a(n) = floor(n*phi) - n, where phi = (1+sqrt(5))/2. - William A. Tedeschi (fynmun(AT)hotmail.com), Mar 06 2008
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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.
From Wolfdieter Lang, Jun 28 2011: (Start)
There are a(4)=2 (positive) Wythoff A-numbers <=3, namely 1 and 3.
There are a(4)=2 (positive) Wythoff B-numbers <A(4)=6, namely 2 and 5.
a(4) = 2 = A(4) - 4 = 6 - 4.
(End)
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PROG
| (PARI) { default(realprecision, 10); p=(sqrt(5) - 1)/2; for (n=0, 1000, write("b060143.txt", n, " ", floor(n*p)); ) } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Jul 02 2009]
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CROSSREFS
| Cf. A074840, A074065, A035336, A022342, A066094-A066096.
Apart from initial terms, same as A005206.
Sequence in context: A090638 A057363 A073869 * A005206 A057365 A014245
Adjacent sequences: A060140 A060141 A060142 * A060144 A060145 A060146
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KEYWORD
| easy,frac,nonn
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AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu), Mar 05 2001
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EXTENSIONS
| I merged three identical sequences to create this entry. Some of the formulae may need their initial terms adjusting now. - N. J. A. Sloane (njas(AT)research.att.com), Mar 05 2003
More terms from William A. Tedeschi (fynmun(AT)hotmail.com), Mar 06 2008
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