A130242 - OEIS

A130242

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

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, 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, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10

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OFFSET

0,4

COMMENTS

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).

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..5000

FORMULA

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

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

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

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

EXAMPLE

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

MATHEMATICA

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

PROG

(Python)

from itertools import islice, count

def A130242_gen(): # generator of terms

yield from (0, 0, 0, 2)

a, b = 3, 4

for i in count(3):

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

a, b = b, a+b

A130242_list = list(islice(A130242_gen(), 40)) # Chai Wah Wu, Jun 08 2022

CROSSREFS

For partial sums see A130244.

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

Indicator sequence A102460.

Fibonacci inverse see A130233 - A130240, A104162.

Sequence in context: A225486 A325282 A305233 * A130245 A087793 A030411

Adjacent sequences: A130239 A130240 A130241 * A130243 A130244 A130245

KEYWORD

nonn

AUTHOR

Hieronymus Fischer, May 19 2007, Jul 02 2007

STATUS

approved