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Minimal index k of an even Fibonacci number A001906 such that A001906(k)=Fib(2k)>=n (the 'upper' even Fibonacci Inverse).

+0
7

0, 1, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6 (list; graph; listen)

	OFFSET	`0,3`
	COMMENT	`Inverse of the even Fibonacci sequence (A001906), since a(A001906(n))=n (see A130259 for another version). a(n+1) is the number of even Fibonacci numbers (A001906) <=n.`
	FORMULA	`a(n)=ceiling(arsinh(sqr(5)n/2)/(2ln(phi))) for n>=0.` `a(n)=ceiling(arcosh(sqr(5)n/2)/(2ln(phi))) for n>=1.` `a(n)=ceiling(log_phi(sqr(5)n)/2)=ceiling(log_phi(sqr(5)n-1)/2) for n>=1, where phi=(1+sqr(5))/2.` `a(n)=A130259(n-1)+1, for n>=1.` `G.f.: g(x)=x/(1-x)*sum(k>=0, x^Fib(2k)).`
	EXAMPLE	`a(10)=4 because A001906(4)=21>=10, but A001906(3)=8<10.`
	CROSSREFS	`Cf. partial sums A130262. Other related sequences: A000045, A001519, A130234, A130237, A130239, A130256, A130259. Lucas inverse: A130241 - A130248.` `Sequence in context: A084320 A120699 A072643 this_sequence A111393 A062537 A097688` `Adjacent sequences: A130257 A130258 A130259 this_sequence A130261 A130262 A130263`
	KEYWORD	`nonn`
	AUTHOR	`Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), May 25 2007, May 28 2007, Jul 02 2007`

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Last modified December 18 21:37 EST 2009. Contains 171024 sequences.

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