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A130242
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Minimal index k of a Lucas number such that Lucas(k)>=n (the 'upper' Lucas (A000032) Inverse).
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9
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0, 0, 0, 2, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 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, 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
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OFFSET
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0,4
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COMMENTS
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Inverse of the Lucas sequence (A000032), nearly, since a(Lucas(n))=n except for n=1 (see A130241 and A130247 for other versions). For n>=2, a(n+1) is equal to the partial sum of the Lucas indicator sequence (see A102460).
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LINKS
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Table of n, a(n) for n=0..95.
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FORMULA
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a(n)=ceiling(log_phi((n+sqr(n^2-4))/2))=ceiling(arcosh(n/2)/ln(phi)) where phi=(1+sqr(5))/2.
a(n)=A130241(n-1)+1=A130245(n-1) for n>=3.
G.f.: g(x)=x/(1-x)*(2x^2+sum{k>=2, x^Lucas(k)}).
a(n)=ceiling(log_phi(n-1/2)) for n>=3, where phi is the golden ratio.
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EXAMPLE
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a(10)=5, since Lucas(5)=11>=10 but Lucas(4)=7<10.
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CROSSREFS
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For partial sums see A130244. Other related sequences: A000032, A130241, A130245, A130247, A130250, A130256, A130260. Indicator sequence A102460. Fibonacci inverse see A130233 - A130240, A104162.
Sequence in context: A136528 A225486 * A130245 A087793 A030411 A194817
Adjacent sequences: A130239 A130240 A130241 * A130243 A130244 A130245
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KEYWORD
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nonn
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AUTHOR
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Hieronymus Fischer, May 19 2007, Jul 02 2007
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STATUS
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approved
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