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A130241 Maximal index k of a Lucas number such that Lucas(k) <= n (the 'lower' Lucas (A000032) Inverse). 24


%S 1,1,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,

%T 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,

%U 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

%N Maximal index k of a Lucas number such that Lucas(k) <= n (the 'lower' Lucas (A000032) Inverse).

%C Inverse of the Lucas sequence (A000032), nearly, since a(Lucas(n))=n for n>=1 (see A130242 and A130247 for other versions). For n>=2, a(n)+1 is equal to the partial sum of the Lucas indicator sequence (see A102460). Identical to A130247 except for n=2.

%H G. C. Greubel, <a href="/A130241/b130241.txt">Table of n, a(n) for n = 1..5000</a>

%F a(n) = floor(log_phi((n+sqrt(n^2+4))/2)) = floor(arcsinh((n+1)/2)/log(phi)) where phi=(1+sqrt(5))/2.

%F a(n) = A130242(n+1) - 1 for n>=2.

%F a(n) = A130247(n) except for n=2.

%F G.f.: g(x) = 1/(1-x) * Sum{k>=1, x^Lucas(k)}.

%F a(n) = floor(log_phi(n+1/2)) for n>=2, where phi is the golden ratio.

%e a(10)=4, since Lucas(4)=7<=10 but Lucas(5)=11>10.

%t Join[{1}, Table[Floor[Log[GoldenRatio, n + 1/2]], {n, 2, 50}]] (* _G. C. Greubel_, Dec 24 2017 *)

%o (PARI) for(n=1,50, print1(floor(log((2*n+1)/2)/log((1+sqrt(5))/2)), ", ")) \\ _G. C. Greubel_, Sep 09 2018

%o (MAGMA) [Floor(Log((2*n+1)/2)/Log((1+Sqrt(5))/2)): n in [2..50]]; // _G. C. Greubel_, Sep 09 2018

%Y For partial sums see A130243. Other related sequences: A000032, A130242, A130245, A130247, A130249, A130255, A130259. Indicator sequence A102460. Fibonacci inverse see A130233 - A130240, A104162.

%K nonn

%O 1,3

%A _Hieronymus Fischer_, May 19 2007, Jul 02 2007

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Last modified December 14 19:27 EST 2019. Contains 329987 sequences. (Running on oeis4.)