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Search: id:A060143
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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; listen)
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OFFSET
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0,5
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COMMENT
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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.
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FORMULA
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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
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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.
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PROGRAM
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(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
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Cf. A074840, A074065, A035336, A022342, A066094-A066096.
Apart from initial terms, same as A005206.
Adjacent sequences: A060140 A060141 A060142 this_sequence A060144 A060145 A060146
Sequence in context: A090638 A057363 A073869 this_sequence A005206 A057365 A014245
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KEYWORD
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easy,frac,nonn,new
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AUTHOR
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Clark Kimberling (ck6(AT)evansville.edu), Mar 05 2001
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EXTENSIONS
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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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