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Search: id:A130234
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| A130234 |
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Minimal index k of a Fibonacci number such that Fib(k)>=n (the 'upper' Fibonacci Inverse). |
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+0
20
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| 0, 1, 3, 4, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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Inverse of the Fibonacci sequence (A000045), nearly, since a(Fib(n))=n except for n=2 (see A130233 for another version). a(n+1) is equal to the partial sum of the Fibonacci indicator sequence (see A104162).
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FORMULA
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a(n)=ceiling(log_phi((sqr(5)*n+sqr(5*n^2-4))/2))=ceiling(arcosh(sqr(5)*n/2)/ln(phi)) where phi=(1+sqr(5))/2. Also true: a(n)=A130233(n-1)+1 for n>0. G.f.: g(x)=x/(1-x)*sum{k>=0, x^Fib(k)}.
a(n)=ceiling(log_phi(sqr(5)*n-1)) for n>=1, where phi is = the golden ratio. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jul 02 2007
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EXAMPLE
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a(10)=7, since Fib(7)=13>=10 but Fib(6)=8<10.
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CROSSREFS
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Partial sums: A130236. Other related sequences: A000045, A130234, A130256, A130260, A104162, A108852, A130256, A130260, Lucas Inverse: A130241 - A130248.
Sequence in context: A120677 A098200 A092405 this_sequence A108852 A119476 A037038
Adjacent sequences: A130231 A130232 A130233 this_sequence A130235 A130236 A130237
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KEYWORD
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nonn
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AUTHOR
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Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), May 17 2007
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