%I #17 Jun 09 2022 02:26:36

%S 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,

%T 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,

%U 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

%N Minimal index k of a Lucas number such that Lucas(k)>=n (the 'upper' Lucas (A000032) Inverse).

%C 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).

%H G. C. Greubel, <a href="/A130242/b130242.txt">Table of n, a(n) for n = 0..5000</a>

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

%F a(n) = A130241(n-1) + 1 = A130245(n-1) for n>=3.

%F G.f.: x/(1-x)*(2x^2+sum{k>=2, x^Lucas(k)}).

%F a(n) = ceiling(log_phi(n-1/2)) for n>=3, where phi is the golden ratio.

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

%t Join[{0, 0, 0}, Table[Ceiling[Log[GoldenRatio, n + 1/2]], {n, 2, 50}]] (* _G. C. Greubel_, Dec 24 2017 *)

%o (Python)

%o from itertools import islice, count

%o def A130242_gen(): # generator of terms

%o yield from (0,0,0,2)

%o a, b = 3, 4

%o for i in count(3):

%o yield from (i,)*(b-a)

%o a, b = b, a+b

%o A130242_list = list(islice(A130242_gen(),40)) # _Chai Wah Wu_, Jun 08 2022

%Y For partial sums see A130244.

%Y Other related sequences: A000032, A130241, A130245, A130247, A130250, A130256, A130260.

%Y Indicator sequence A102460.

%Y Fibonacci inverse see A130233 - A130240, A104162.

%K nonn

%O 0,4

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