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A130247
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Inverse Lucas (A000032) numbers: index k of a Lucas number such that Lucas(k)=n; max(k|Lucas(k)<n), if there is no such index.
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5
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1, 0, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 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, 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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| Inverse of the Lucas sequence (A000032), since a(Lucas(n))=n for n>=0 (see A130241 and A130242 for other versions). Same as A130241 except for n=1.
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FORMULA
| a(n)=c(n), if (n^2-4)/5 is a square number, a(n)=s(n), if (n^2+4)/5 is a square number and a(n)=floor(log_phi(n)) else, where s(n)=floor(arsinh(n/2)/ln(phi)), c(n)=floor(arcosh(n/2)/ln(phi)) and phi=(1+sqr(5))/2.
a(n)=A130241(n) except for n=2.
G.f.: g(x)=1/(1-x)*sum{k>=1, x^Lucas(k)}-x^2.
a(n)=floor(log_phi(n+1/2)) for n>=3, where phi is the golden ratio.
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EXAMPLE
| a(2)=0, since Lucas(0)=2; a(10)=4, since Lucas(4)=7<10 but Lucas(5)=11>10.
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CROSSREFS
| For partial sums see A130248. Other related sequences: A000032, A130241, A130242, A130245, A130249, A130255, A130259. Indicator sequence A102460. Fibonacci inverse see A130233 - A130240, A104162.
Sequence in context: A117806 A085423 A130241 * A087839 A106742 A106733
Adjacent sequences: A130244 A130245 A130246 * A130248 A130249 A130250
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KEYWORD
| nonn
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AUTHOR
| Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), May 19 2007, Jul 02 2007
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