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Maximal index k of an odd Fibonacci number (A001519) such that A001519(k)=Fib(2k-1)<=n (the 'lower' odd Fibonacci Inverse).

+0
12

1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 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, 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 (list; graph; listen)

	OFFSET	`1,2`
	COMMENT	`Inverse of the odd Fibonacci sequence (A001519), nearly, since a(A001519(n))=n except for n=0 (see A130256 for another version). a(n)+1 is the number of odd Fibonacci numbers (A001519) <=n (for n>=1).`
	FORMULA	`a(n)=floor((1+arsinh(sqr(5)n/2)/ln(phi))/2).` `a(n)=floor((1+arcosh(sqr(5)n/2)/ln(phi))/2).` `a(n)=floor((1+log_phi(sqr(5)n))/2) for n>=1, where phi=(1+sqr(5)/2.` `G.f.: g(x)=1/(1-x)sum(k>=1, x^Fib(2k-1)).` `a(n)=floor(1/2(1+log_phi(sqr(5)n+1))) for n>=1.`
	EXAMPLE	`a(10)=3 because A001519(3)=5<=10, but A001519(4)=13>10.`
	CROSSREFS	`Cf. partial sums A130257. Other related sequences: A000045, A130233, A130237, A130239, A130256, A130259, A104160. Lucas inverse: A130241 - A130248.` `Sequence in context: A101787 A064099 A134021 this_sequence A082527 A132944 A106432` `Adjacent sequences: A130252 A130253 A130254 this_sequence A130256 A130257 A130258`
	KEYWORD	`nonn`
	AUTHOR	`Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), May 24 2007, Jul 02 2007`

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Last modified November 25 20:09 EST 2009. Contains 167514 sequences.

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