A130241 -id:A130241 - cp's OEIS frontend

A130243 Partial sums of the 'lower' Lucas Inverse A130241.

1, 2, 4, 7, 10, 13, 17, 21, 25, 29, 34, 39, 44, 49, 54, 59, 64, 70, 76, 82, 88, 94, 100, 106, 112, 118, 124, 130, 137, 144, 151, 158, 165, 172, 179, 186, 193, 200, 207, 214, 221, 228, 235, 242, 249, 256, 264, 272, 280, 288, 296, 304, 312, 320, 328, 336, 344, 352

Offset: 1

Hieronymus Fischer, May 19 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 1..2500

Other related sequences: A000032, A130244, A130242, A130245, A130246, A130248, A130251, A130257, A130261. Fibonacci inverse see A130233 - A130240, A104162.

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

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

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

a(n) = Sum_{k=1..n} A130241(k).
a(n) = (n+1)*A130241(n) - A000032(A130241(n)+2) + 3.
G.f.: g(x) = 1/(1-x)^2*Sum_{k>=1} x^Lucas(k).

A130233 a(n) is the maximal k such that Fibonacci(k) <= n (the "lower" Fibonacci Inverse).

Original entry on oeis.org

0, 2, 3, 4, 4, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 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, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10

Offset: 0

Hieronymus Fischer, May 17 2007

Inverse of the Fibonacci sequence (A000045), nearly, since a(Fibonacci(n)) = n except for n = 1 (see A130234 for another version). a(n) + 1 is equal to the partial sum of the Fibonacci indicator sequence (see A104162).

			a(10) = 6, since Fibonacci(6) = 8 <= 10 but Fibonacci(7) = 13 > 10.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Cf. A130235 (partial sums), A104162 (first differences).
Other related sequences: A000045, A130234, A130237, A130239, A130255, A130259, A108852. Lucas inverse: A130241.
Cf. A001622 (golden ratio), A002390 (its log).

fibLLog[0] := 0; fibLLog[1] := 2; fibLLog[n_Integer] := fibLLog[n] = If[n < Fibonacci[fibLLog[n - 1] + 1], fibLLog[n - 1], fibLLog[n - 1] + 1]; Table[fibLLog[n], {n, 0, 88}] (* Alonso del Arte, Sep 01 2013 *)

a(n)=log(sqrt(5)*n+1.5)\log((1+sqrt(5))/2) \\ Charles R Greathouse IV, Mar 21 2012

a(n) = floor(log_phi((sqrt(5)*n + sqrt(5*n^2+4))/2)) where phi = (1+sqrt(5))/2 = A001622.
a(n) = floor(arcsinh(sqrt(5)*n/2) / log(phi)), with log(phi) = A002390.
a(n) = A130234(n+1) - 1.
G.f.: g(x) = 1/(1-x) * Sum_{k>=1} x^Fibonacci(k).
a(n) = floor(log_phi(sqrt(5)*n+1)), n >= 0, where phi is the golden ratio. - Hieronymus Fischer, Jul 02 2007

A130234 Minimal index k of a Fibonacci number such that Fibonacci(k) >= n (the 'upper' Fibonacci Inverse).

Original entry on oeis.org

0, 1, 3, 4, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 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, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11

Offset: 0

Hieronymus Fischer, May 17 2007

Inverse of the Fibonacci sequence (A000045), nearly, since a(Fibonacci(n)) = n except for n = 2 (see A130233 for another version). a(n+1) is equal to the partial sum of the Fibonacci indicator sequence (see A104162).

			a(10) = 7, since Fibonacci(7) = 13 >= 10 but Fibonacci(6) = 8 < 10.

Partial sums: A130236.
Other related sequences: A000045, A130233, A130256, A130260, A104162, A108852.
Lucas inverse: A130241 - A130248.

A130234 := proc(n)
    local i;
    for i from 0 do
        if A000045(i) >= n then
            return i;
        end if;
    end do:
end proc: # R. J. Mathar, Jan 31 2015

a[n_] := For[i = 0, True, i++, If[Fibonacci[i] >= n, Return[i]]];
a /@ Range[0, 80] (* Jean-François Alcover, Apr 13 2020 *)

a(n)=my(k);while(fibonacci(k)Charles R Greathouse IV, Feb 03 2014, corrected by M. F. Hasler, Apr 07 2021

a(n) = ceiling(log_phi((sqrt(5)*n + sqrt(5*n^2-4))/2)) = ceiling(arccosh(sqrt(5)*n/2)/log(phi)) where phi = (1+sqrt(5))/2, the golden ratio, for n > 0.
a(n) = A130233(n-1) + 1 for n > 0.
G.f.: x/(1-x) * Sum_{k >= 0} x^Fibonacci(k).
a(n) = ceiling(log_phi(sqrt(5)*n - 1)) for n > 0, where phi is the golden ratio. - Hieronymus Fischer, Jul 02 2007
a(n) = A108852(n-1). - R. J. Mathar, Jan 31 2015

A130248 Partial sums of the Lucas Inverse A130247.

Original entry on oeis.org

1, 1, 3, 6, 9, 12, 16, 20, 24, 28, 33, 38, 43, 48, 53, 58, 63, 69, 75, 81, 87, 93, 99, 105, 111, 117, 123, 129, 136, 143, 150, 157, 164, 171, 178, 185, 192, 199, 206, 213, 220, 227, 234, 241, 248, 255, 263, 271, 279, 287, 295, 303, 311, 319, 327, 335, 343, 351

Offset: 1

Hieronymus Fischer, May 19 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 1..5000

Other related sequences: A000032, A130241, A130242, A130243, A130244, A130245, A130246, A130251, A130252, A130257, A130261. Fibonacci inverse see A130233 - A130240, A104162.

Join[{1, 1}, Table[Sum[Floor[Log[GoldenRatio, k + 1/2]], {k, 1, n}], {n, 3, 50}]] (* G. C. Greubel, Dec 24 2017 *)

a(n)=sum{1<=k<=n, A130247(k)}=2+(n+1)*A130247(n)-A000032(A130247(n)+2) for n>=3. G.f.: g(x)=1/(1-x)^2*(sum{k>=1, x^Lucas(k)}-x^2).

A130249 Maximal index k of a Jacobsthal number such that A001045(k)<=n (the 'lower' Jacobsthal inverse).

Original entry on oeis.org

0, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 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, 7, 7, 7, 7, 7, 7, 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

Offset: 0

Hieronymus Fischer, May 20 2007

nonn

Inverse of the Jacobsthal sequence (A001045), nearly, since a(A001045(n))=n except for n=1 (see A130250 for another version). a(n)+1 is equal to the partial sum of the Jacobsthal indicator sequence (see A105348).

			a(12)=5, since A001045(5)=11<=12, but A001045(6)=21>12.

G. C. Greubel, Table of n, a(n) for n = 0..10000

For partial sums see A130251.
Other related sequences A130250, A130253, A105348, A001045, A130233, A130241.
Cf. A000523, A078008 (runlengths).

[Floor(Log(3*n+1)/Log(2)): n in [0..30]]; // G. C. Greubel, Jan 08 2018

Table[Floor[Log[2, 3*n + 1]], {n, 0, 50}] (* G. C. Greubel, Jan 08 2018 *)

for(n=0, 30, print1(floor(log(3*n+1)/log(2)), ", ")) \\ G. C. Greubel, Jan 08 2018

a(n) = logint(3*n+1, 2); \\ Ruud H.G. van Tol, May 12 2024

def A130249(n): return (3*n+1).bit_length()-1 # Chai Wah Wu, Jun 08 2022

a(n) = floor(log_2(3n+1)).
a(n) = A130250(n+1) - 1 = A130253(n) - 1.
G.f.: 1/(1-x)*(Sum_{k>=1} x^A001045(k)).
a(n) = A000523(3*n+1). - Ruud H.G. van Tol, May 12 2024

A102460 a(n) = 1 if n is a Lucas number, else a(n) = 0.

Original entry on oeis.org

0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Casey Mongoven, Apr 18 2005

nonn

From Hieronymus Fischer, Jul 02 2007: (Start)
a(n) is the number of nonnegative integer solutions to 25*x^4-10*n^2*x^2+n^4-16=0.
a(n)=1 if and only if there is an integer m such that x=n is a root of p(x)=x^4-10*m^2*x^2+25*m^4-16.
For n>=3: a(n)=1 iff floor(log_phi(n+1/2))=ceiling(log_phi(n-1/2)). (End)

Antti Karttunen, Table of n, a(n) for n = 0..65537
Casey Mongoven, Lucas Binary no. 1; electronic music created with this sequence.
Index entries for characteristic functions

Cf. A000032, A130241, A130242, A130247, A123927.
Cf. partial sums A130245.
Cf. also A010056, A104162, A105348, A147612, A294878.

{0}~Join~ReplacePart[ConstantArray[0, Last@ #], Map[# -> 1 &, #]] &@ Array[LucasL, 11, 0] (* Michael De Vlieger, Nov 22 2017 *)
With[{nn=130,lc=LucasL[Range[0,20]]},Table[If[MemberQ[lc,n],1,0],{n,0,nn}]] (* Harvey P. Dale, Jul 03 2022 *)

a(n)=my(f=factor(25*'x^4-10*n^2*'x^2+n^4-16)[,1]); sum(i=1,#f, poldegree(f[i])==1 && polcoeff(f[i],0)<=0) \\ Charles R Greathouse IV, Nov 06 2014

A102460(n) = { my(u1=1,u2=3,old_u1); if(n<=2,sign(n),while(n>u2,old_u1=u1;u1=u2;u2=old_u1+u2);(u2==n)); }; \\ Antti Karttunen, Nov 22 2017

from sympy.ntheory.primetest import is_square
def A102460(n): return int(is_square(m:=5*(n**2-4)) or is_square(m+40)) # Chai Wah Wu, Jun 13 2024

From Hieronymus Fischer, Jul 02 2007: (Start)
G.f.: g(x) = Sum_{k>=0} x^A000032(k).
a(n) = 1+floor(arcsinh(n/2)/log(phi))-ceiling(arccosh(n/2)/log(phi)) for n>=3, where phi=(1+sqrt(5))/2.
a(n) = 1+A130241(n)-A130242(n) for n>=3.
a(n) = 1+A130247(n)-A130242(n) for n=>2.
a(n) = A130245(n)-A130245(n-1) for n>=1.
For n>=3: a(n)=1 iff A130241(n)=A130242(n). (End)

Data section extended up to a(123) by Antti Karttunen, Nov 22 2017

A130245 Number of Lucas numbers (A000032) <= n.

Original entry on oeis.org

0, 1, 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

Offset: 0

Hieronymus Fischer, May 19 2007, Jul 02 2007

nonn

Partial sums of the Lucas indicator sequence A102460.

For n>=2, we have a(A000032(n)) = n + 1.

			a(9)=5 because there are 5 Lucas numbers <=9 (2,1,3,4 and 7).

Antti Karttunen, Table of n, a(n) for n = 0..64079
Dorin Andrica, Ovidiu Bagdasar, and George Cătălin Tųrcąs, On some new results for the generalised Lucas sequences, An. Şt. Univ. Ovidius Constanţa (Romania, 2021) Vol. 29, No. 1, 17-36.

Partial sums of A102460.
For partial sums of this sequence, see A130246. Other related sequences: A000032, A130241, A130242, A130247, A130249, A130253, A130255, A130259.
For Fibonacci inverse, see A130233 - A130240, A104162, A108852.

[0] cat [1+Floor(Log((2*n+1)/2)/Log((1+Sqrt(5))/2)): n in [1..100]]; // G. C. Greubel, Sep 09 2018

Join[{0}, Table[1+Floor[Log[GoldenRatio, (2*n+1)/2]], {n,1,100}]] (* G. C. Greubel, Sep 09 2018 *)

A102460(n) = { my(u1=1,u2=3,old_u1); if(n<=2,sign(n),while(n>u2,old_u1=u1;u1=u2;u2=old_u1+u2);(u2==n)); };
A130245(n) = if(!n,n,A102460(n)+A130245(n-1));
\\ Or just as:
c=0; for(n=0,123,c += A102460(n); print1(c,", ")); \\ Antti Karttunen, May 13 2018

from itertools import count, islice
def A130245_gen(): # generator of terms
    yield from (0, 1, 2)
    a, b = 3,4
    for i in count(3):
        yield from (i,)*(b-a)
        a, b = b, a+b
A130245_list = list(islice(A130245_gen(),40)) # Chai Wah Wu, Jun 08 2022

a(n) = 1 +floor(log_phi((n+sqrt(n^2+4))/2)) = 1 +floor(arcsinh(n/2)/log(phi)) for n>=2, where phi = (1+sqrt(5))/2.
a(n) = A130241(n)+1 = A130242(n+1) for n>=2.
G.f.: g(x) = 1/(1-x)*sum{k>=0, x^Lucas(k)}.
a(n) = 1 +floor(log_phi(n+1/2)) for n>=1, where phi is the golden ratio.
Sum_{n>=1} (-1)^(n+1)/a(n) = 3/2 - Pi/(6*sqrt(3)) - log(3)/2. - Amiram Eldar, Jul 25 2025

A130253 Number of Jacobsthal numbers (A001045) <=n.

Original entry on oeis.org

1, 3, 3, 4, 4, 5, 5, 5, 5, 5, 5, 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, 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, 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

Offset: 0

Hieronymus Fischer, May 20 2007

Partial sums of the Jacobsthal indicator sequence (A105348).

For n<>1, we have a(A001045(n))=n+1.

			a(9)=5 because there are 5 Jacobsthal numbers <=9 (0,1,1,3 and 5).

G. C. Greubel, Table of n, a(n) for n = 0..10000
Dorin Andrica, Ovidiu Bagdasar, and George Cătălin Tųrcąs, On some new results for the generalised Lucas sequences, An. Şt. Univ. Ovidius Constanţa (Romania, 2021) Vol. 29, No. 1, 17-36.

For partial sums see A130252. Other related sequences A001045, A130249, A130250, A130253, A105348. Also A130233, A130235, A130241, A108852, A130245.

[Ceiling(Log(3*n+2)/Log(2)): n in [0..30]]; // G. C. Greubel, Jan 08 2018

Table[1+Floor[Log[2,3n+1]],{n,0,100}] (* Harvey P. Dale, Jul 03 2013 *)

a(n)=logint(3*n+1,2)+1 \\ Charles R Greathouse IV, Oct 03 2016

def A130253(n): return (3*n+1).bit_length() # Chai Wah Wu, Apr 17 2025

a(n) = floor(log_2(3n+1)) + 1 = ceiling(log_2(3n+2)).
a(n) = A130249(n) + 1 = A130250(n+1).
G.f.: 1/(1-x)*(Sum_{k>=0} x^A001045(k)).

A130255 Maximal index k of an odd Fibonacci number (A001519) such that A001519(k) = Fibonacci(2k-1) <= n (the 'lower' odd Fibonacci Inverse).

Original entry on oeis.org

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

Offset: 1

Hieronymus Fischer, May 24 2007, Jul 02 2007

nonn

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).

			a(10)=3 because A001519(3) = 5 <= 10, but A001519(4) = 13 > 10.

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

Cf. partial sums A130257. Other related sequences: A000045, A130233, A130237, A130239, A130256, A130259, A104160. Lucas inverse: A130241 - A130248.

phi:=(1+Sqrt(5))/2; [Floor((1 +Argsinh(Sqrt(5)*n/2)/Log(phi))/2): n in [1..100]]; // G. C. Greubel, Sep 09 2018

Table[Floor[(1 +ArcSinh[Sqrt[5]*n/2]/Log[GoldenRatio])/2], {n, 1, 100}] (* G. C. Greubel, Sep 09 2018 *)

phi=(1+sqrt(5))/2; vector(100, n, floor((1 +asinh(sqrt(5)*n/2)/log(phi))/2)) \\ G. C. Greubel, Sep 09 2018

a(n) = floor((1 + arcsinh(sqrt(5)*n/2)/log(phi))/2).
a(n) = floor((1 + arccosh(sqrt(5)*n/2)/log(phi))/2).
a(n) = floor((1 + log_phi(sqrt(5)*n))/2) for n >= 1, where phi = (1 + sqrt(5))/2.
G.f.: g(x) = 1/(1-x)*Sum_{k>=1} x^Fibonacci(2k-1).
a(n) = floor((1/2)*(1 + log_phi(sqrt(5)*n + 1))) for n >= 1.

A130257 Partial sums of the 'lower' odd Fibonacci Inverse A130255.

Original entry on oeis.org

1, 3, 5, 7, 10, 13, 16, 19, 22, 25, 28, 31, 35, 39, 43, 47, 51, 55, 59, 63, 67, 71, 75, 79, 83, 87, 91, 95, 99, 103, 107, 111, 115, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 170, 175, 180, 185, 190, 195, 200, 205, 210, 215, 220, 225, 230, 235, 240, 245, 250

Offset: 1

Hieronymus Fischer, May 24 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000045, A001519, A001906, A130233, A130235, A130236, A130256, A130258, A104161. Lucas inverse: A130241 - A130248.

[(&+[Floor((1+Argsinh(Sqrt(5)*k/2)/Log((1+Sqrt(5))/2))/2): k in [1..n]]): n in [1..100]]; // G. C. Greubel, Sep 09 2018

Table[Sum[Floor[(1 + ArcSinh[Sqrt[5]*k/2]/Log[GoldenRatio])/2], {k, 1, n}], {n, 1, 100}] (* G. C. Greubel, Sep 09 2018 *)

for(n=1,100, print1(sum(k=1,n, floor((1+asinh(sqrt(5)*k/2)/log((1+sqrt(5))/2))/2)), ", ")) \\ G. C. Greubel, Sep 09 2018

a(n) = (n+1)*A130255(n) - A001906(A130255(n)).
a(n) = (n+1)*A130255(n) - Fib(2*A130255(n)).
G.f.: g(x)=1/(1-x)^2*sum(k>=1, x^Fib(2k-1)).

cp's OEIS Frontend

A130243 Partial sums of the 'lower' Lucas Inverse A130241.

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A130233 a(n) is the maximal k such that Fibonacci(k) <= n (the "lower" Fibonacci Inverse).

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A130234 Minimal index k of a Fibonacci number such that Fibonacci(k) >= n (the 'upper' Fibonacci Inverse).

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A130248 Partial sums of the Lucas Inverse A130247.

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A130249 Maximal index k of a Jacobsthal number such that A001045(k)<=n (the 'lower' Jacobsthal inverse).

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A102460 a(n) = 1 if n is a Lucas number, else a(n) = 0.

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A130245 Number of Lucas numbers (A000032) <= n.

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