A130241 -id:A130241 - cp's OEIS frontend

A130247 Inverse Lucas (A000032) numbers: index k of a Lucas number such that Lucas(k)=n; max(k|Lucas(k) < n), if there is no such index.

1, 0, 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, 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, 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

Offset: 1

Hieronymus Fischer, May 19 2007, Jul 02 2007

nonn

Inverse of the Lucas sequence (A000032), since a(Lucas(n))=n for n >= 0 (see A130241 and A130242 for other versions). Same as A130241 except for n=1.

			a(2)=0, since Lucas(0)=2; a(10)=4, since Lucas(4) = 7 < 10 but Lucas(5) = 11 > 10.

G. C. Greubel, Table of n, a(n) for n = 1..10000

For partial sums see A130248. Other related sequences: A000032, A130241, A130242, A130245, A130249, A130255, A130259. Indicator sequence A102460. For Fibonacci inverse see A130233 - A130240, A104162.

Join[{1, 0}, Table[Floor[Log[GoldenRatio, n + 1/2]], {n, 3, 50}]] (* G. C. Greubel, Dec 21 2017 *)

from itertools import islice, count
def A130247_gen(): # generator of terms
    yield from (1,0)
    a, b = 3, 4
    for i in count(2):
        yield from (i,)*(b-a)
        a, b = b, a+b
A130247_list = list(islice(A130247_gen(),40)) # Chai Wah Wu, Jun 08 2022

a(n)=c(n), if (n^2-4)/5 is a square number, a(n)=s(n), if (n^2+4)/5 is a square number and a(n)=floor(log_phi(n)) otherwise, where s(n)=floor(arcsinh(n/2)/log(phi)), c(n)=floor(arccosh(n/2)/log(phi)) and phi=(1+sqrt(5))/2.
a(n) = A130241(n) except for n=2.
G.f.: g(x) = (1/(1-x))*(Sum_{k>=1} x^Lucas(k)) - x^2.
a(n) = floor(log_phi(n+1/2)) for n >= 3, where phi is the golden ratio.

A130258 Partial sums of the 'upper' odd Fibonacci Inverse A130256.

Original entry on oeis.org

0, 0, 2, 5, 8, 11, 15, 19, 23, 27, 31, 35, 39, 43, 48, 53, 58, 63, 68, 73, 78, 83, 88, 93, 98, 103, 108, 113, 118, 123, 128, 133, 138, 143, 148, 154, 160, 166, 172, 178, 184, 190, 196, 202, 208, 214, 220, 226, 232, 238, 244, 250, 256, 262, 268, 274, 280, 286, 292

Offset: 0

Hieronymus Fischer, May 24 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 0..2500

Cf. A000045, A001519, A001906, A130234, A130235, A130236, A130255, A130257, A104162. Lucas inverse: A130241 - A130248.

[0,0] cat [(&+[Ceiling((1/2)*(1 + Log(Sqrt(5)*k-1)/Log((1+Sqrt(5))/2))): k in [2..n]]): n in [2..50]]; // G. C. Greubel, Sep 13 2018

Table[Sum[Ceiling[1/2*(1 + Log[GoldenRatio, (Sqrt[5]*k - 1)])], {k,2,n}], {n, 0, 50}] (* G. C. Greubel, Sep 13 2018 *)

for(n=0, 50, print1(if(n==0, 0, if(n==1, 0, sum(k=2, n, ceil( (1/2)*(1 + log(sqrt(5)*k - 1)/log((1+sqrt(5))/2)))))), ", ")) \\ G. C. Greubel, Sep 13 2018

a(n) = n*A130256(n) - A001906(A130256(n) -1).
a(n) = n*A130256(n) - Fib(2*A130256(n)-2) - 1.
G.f.: g(x) = x/(1-x)^2*Sum_{k>=0} x^Fib(2*k-1).

A130262 Partial sums of the 'upper' even Fibonacci Inverse A130260.

Original entry on oeis.org

0, 1, 3, 5, 8, 11, 14, 17, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 77, 82, 87, 92, 97, 102, 107, 112, 117, 122, 127, 132, 137, 142, 147, 152, 157, 162, 167, 172, 177, 182, 187, 192, 197, 202, 207, 212, 217, 222, 227, 232, 237, 242, 248, 254, 260, 266

Offset: 0

Hieronymus Fischer, May 25 2007

nonn

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

Cf. A000045, A001519, A001906, A130234, A130235, A130236, A130256, A130259, A104161. Lucas inverse: A130241 - A130248.

[0] cat [(&+[ Ceiling(Log(Sqrt(5)*k)/(2*Log((1+ Sqrt(5))/2))): k in [1..n]]): n in [1..50]]; // G. C. Greubel, Sep 12 2018

Table[Sum[Ceiling[Log[GoldenRatio, Sqrt[5]*k]/2], {k, 1, n}], {n, 0, 60}] (* G. C. Greubel, Sep 12 2018 *)

for(n=0, 50, print1(sum(k=1,n, ceil(log(sqrt(5)*k)/(2*log((1+ sqrt(5))/2)))), ", ")) \\ G. C. Greubel, Sep 12 2018

a(n) = n*A130260(n) - A001519(A130260(n)) + 1.
a(n) = n*A130260(n) - Fib(2*A130260(n)-1) + 1.
G.f.: g(x)=x/(1-x)^2*Sum_{k>=0} x^Fib(2*k).

A352717 Greatest Lucas number that does not exceed n.

Original entry on oeis.org

1, 1, 3, 4, 4, 4, 7, 7, 7, 7, 11, 11, 11, 11, 11, 11, 11, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47

Offset: 1

Clark Kimberling, Apr 01 2022

nonn

			The Lucas numbers, beginning with 1, are 1, 3, 4, 7, 11, 18, ..., so that a(5) = 4.

Cf. A000032, A000204, A087172, A130241, A352718.

Flatten[Map[ConstantArray[LucasL[#], LucasL[# - 1]] &, Range[15]]] (* Peter J. C. Moses, Apr 30 2022 *)

from itertools import islice
def A352717_gen(): # generator of terms
    a, b = 1, 3
    while True:
        yield from (a,)*(b-a)
        a, b = b, a+b
A352717_list = list(islice(A352717_gen(),40)) # Chai Wah Wu, Jun 08 2022

cp's OEIS Frontend

A130247 Inverse Lucas (A000032) numbers: index k of a Lucas number such that Lucas(k)=n; max(k|Lucas(k) < n), if there is no such index.

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A130258 Partial sums of the 'upper' odd Fibonacci Inverse A130256.

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A130262 Partial sums of the 'upper' even Fibonacci Inverse A130260.

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A352717 Greatest Lucas number that does not exceed n.

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