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Search: id:A130233
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| A130233 |
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Maximal index k of a Fibonacci number such that Fib(k)<=n (the 'lower' Fibonacci Inverse). |
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+0
32
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| 0, 2, 3, 4, 4, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 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, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Inverse of the Fibonacci sequence (A000045), nearly, since a(Fib(n))=n except for n=1 (see A130234 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)=floor(log_phi((sqr(5)*n+sqr(5*n^2+4))/2))=floor(arsinh(sqr(5)*n/2)/ln(phi)) where phi=(1+sqr(5))/2. Also true: a(n)=A130234(n+1)-1. G.f.: g(x)=1/(1-x)*sum{k>=1, x^Fib(k)}.
a(n)=floor(log_phi(sqr(5)*n+1)), n>=0, 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)=6, since Fib(6)=8<=10 but Fib(7)=13>10.
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CROSSREFS
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Partial sums: A130235. Other related sequences: A000045, A130234, A130237, A130239, A130255, A130259, A104162, A108852, A130255, A130259. Lucas inverse: A130241 - A130248.
Adjacent sequences: A130230 A130231 A130232 this_sequence A130234 A130235 A130236
Sequence in context: A092338 A030601 A049839 this_sequence A131234 A056791 A027434
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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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