A130240 -id:A130240 - cp's OEIS frontend

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

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, 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), 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.

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

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

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

[Floor(Log((2*n+1)/2)/Log((1+Sqrt(5))/2)): n in [2..50]]; // G. C. Greubel, Sep 09 2018

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

for(n=1,50, print1(floor(log((2*n+1)/2)/log((1+sqrt(5))/2)), ", ")) \\ G. C. Greubel, Sep 09 2018

from itertools import count, islice
def A130241_gen(): # generator of terms
    a, b = 1, 3
    for i in count(1):
        yield from (i,)*(b-a)
        a, b = b, a+b
A130241_list = list(islice(A130241_gen(),40)) # Chai Wah Wu, Jun 08 2022

a(n) = floor(log_phi((n+sqrt(n^2+4))/2)) = floor(arcsinh((n+1)/2)/log(phi)) where phi=(1+sqrt(5))/2.
a(n) = A130242(n+1) - 1 for n>=2.
a(n) = A130247(n) except for n=2.
G.f.: g(x) = 1/(1-x) * Sum{k>=1, x^Lucas(k)}.
a(n) = floor(log_phi(n+1/2)) for n>=2, where phi is the golden ratio.

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).

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

A130235 Partial sums of the 'lower' Fibonacci Inverse A130233.

Original entry on oeis.org

0, 2, 5, 9, 13, 18, 23, 28, 34, 40, 46, 52, 58, 65, 72, 79, 86, 93, 100, 107, 114, 122, 130, 138, 146, 154, 162, 170, 178, 186, 194, 202, 210, 218, 227, 236, 245, 254, 263, 272, 281, 290, 299, 308, 317, 326, 335, 344, 353, 362, 371, 380, 389, 398, 407, 417, 427

Offset: 0

Hieronymus Fischer, May 17 2007

nonn

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

Cf. A000045, A130233, A130234, A130236, A130238, A130240, A130243, A130246, A130244, A130246, A130248, A130251, A130257, A130261.

m:=120;
f:= func< x | (&+[x^Fibonacci(j): j in [1..Floor(3*Log(3*m+1))]])/(1-x)^2 >;
R:=PowerSeriesRing(Rationals(), m+1);
[0] cat Coefficients(R!( f(x) )); // G. C. Greubel, Mar 17 2023

nmax = 90; CoefficientList[Series[Sum[x^Fibonacci[k], {k, 1, 1 + Log[3/2 + Sqrt[5]*nmax]/Log[GoldenRatio]}]/(1-x)^2, {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 14 2020 *)

m=120
def f(x): return sum( x^fibonacci(j) for j in range(1, int(3*log(3*m+1))))/(1-x)^2
def A130235_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( f(x) ).list()
A130235_list(m) # G. C. Greubel, Mar 17 2023

a(n) = Sum_{k=0..n} A130233(k) = (n+1)*A130233(n) - Fib(A130233(n)+2) + 1.

G.f.: 1/(1-x)^2 * Sum_{k>=1} x^Fib(k). [corrected by Joerg Arndt, Apr 14 2020]

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

Original entry on oeis.org

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).

A130246 Partial sums of A130245.

Original entry on oeis.org

0, 1, 3, 6, 10, 14, 18, 23, 28, 33, 38, 44, 50, 56, 62, 68, 74, 80, 87, 94, 101, 108, 115, 122, 129, 136, 143, 150, 157, 165, 173, 181, 189, 197, 205, 213, 221, 229, 237, 245, 253, 261, 269, 277, 285, 293, 301, 310, 319, 328, 337, 346, 355, 364, 373, 382, 391

Offset: 0

Hieronymus Fischer, May 19 2007

nonn

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

Other related sequences: A000032, A130241, A130243, A130244, A130248, A130251, A130252, A130255, A130257, A130261. Fibonacci inverse see A130233 - A130240, A104162.

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

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

for(n=0, 100, print1(sum(k=1,n, 1 + floor(log((2*k+1)/2)/log((1+sqrt(5))/2))), ", ")) \\ G. C. Greubel, Sep 09 2018

a(n) = Sum_{k=1..n} A130245(k).
a(n) = 1 +(n+1)*A130245(n) - A000032(A130245(n)+1) for n=0 or n >= 2.
G.f.: 1/(1-x)^2*Sum_{k>=0} x^A000032(k).

A130242 Minimal index k of a Lucas number such that Lucas(k)>=n (the 'upper' Lucas (A000032) Inverse).

Original entry on oeis.org

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

Offset: 0

Hieronymus Fischer, May 19 2007, Jul 02 2007

nonn

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).

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

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

For partial sums see A130244.
Other related sequences: A000032, A130241, A130245, A130247, A130250, A130256, A130260.
Indicator sequence A102460.
Fibonacci inverse see A130233 - A130240, A104162.

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

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

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.

A130237 The 'lower' Fibonacci Inverse A130233(n) multiplied by n.

Original entry on oeis.org

0, 2, 6, 12, 16, 25, 30, 35, 48, 54, 60, 66, 72, 91, 98, 105, 112, 119, 126, 133, 140, 168, 176, 184, 192, 200, 208, 216, 224, 232, 240, 248, 256, 264, 306, 315, 324, 333, 342, 351, 360, 369, 378, 387, 396, 405, 414, 423, 432, 441, 450, 459, 468, 477, 486, 550

Offset: 0

Hieronymus Fischer, May 17 2007

nonn

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

Partial sums: A130238.

Cf. A000045, A130233, A130234, A130235, A130236, A130238, A130239, A130240, A130243, A130246, A130248, A130239, A130251, A130253, A130257, A130261.

[n*Floor(Log(3/2 +n*Sqrt(5))/Log((1+Sqrt(5))/2)): n in [0..70]]; // G. C. Greubel, Mar 18 2023

Table[n*Floor[Log[GoldenRatio, 3/2 +n*Sqrt[5]]], {n,0,70}] (* G. C. Greubel, Mar 18 2023 *)

[n*int(log(3/2 +n*sqrt(5), golden_ratio)) for n in range(71)] # G. C. Greubel, Mar 18 2023

a(n) = n*A130233(n).
a(n) = n*floor(arcsinh(sqrt(5)*n/2)/log(phi)).
G.f.: (1/(1-x))*Sum_{k>=1} (Fib(k) + x/(1-x))*x^Fib(k).

A130239 Maximal index k of the square of a Fibonacci number such that Fib(k)^2 <= n (the 'lower' squared Fibonacci Inverse).

Original entry on oeis.org

0, 2, 2, 2, 3, 3, 3, 3, 3, 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, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 0

Hieronymus Fischer, May 17 2007, May 28 2007

nonn

			a(10) = 4 since Fib(4)^2 = 9 <= 10 but Fib(5)^2 = 25 > 10.

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

Partial sums: A130240. Other related sequences: A000045, A130233, A130234, A130235, A130236, A130237, A130238, A130240, A130243, A130246, A130248, A130239, A130251, A130253, A130257, A130261.

A130233:= func< n | Floor(Log(3/2 + n*Sqrt(5))/Log((1+Sqrt(5))/2)) >;
[A130233(Floor(Sqrt(n))): n in [0..120]]; // G. C. Greubel, Mar 18 2023

A130233[n_]:= Floor[Log[GoldenRatio, 3/2 +n*Sqrt[5]]];
Table[A130233[Floor[Sqrt[n]]], {n, 0, 120}] (* G. C. Greubel, Mar 18 2023 *)

def A130233(n): return int(log(3/2 +n*sqrt(5), golden_ratio))
def A130239(n): return A130233(floor(sqrt(n)))
[A130239(n) for n in range(121)] # G. C. Greubel, Mar 18 2023

a(n) = max(k | Fib(k)^2 <= n) = A130233(floor(sqrt(n))).
a(n) = floor(arcsinh(sqrt(5n)/2)/log(phi)), where phi=(1+sqrt(5))/2.
G.f.: (1/(1-x))*Sum_{k>=1} x^(Fib(k)^2).

A130244 Partial sums of the 'upper' Lucas Inverse A130242.

Original entry on oeis.org

0, 0, 0, 2, 5, 9, 13, 17, 22, 27, 32, 37, 43, 49, 55, 61, 67, 73, 79, 86, 93, 100, 107, 114, 121, 128, 135, 142, 149, 156, 164, 172, 180, 188, 196, 204, 212, 220, 228, 236, 244, 252, 260, 268, 276, 284, 292, 300, 309, 318, 327, 336, 345, 354, 363, 372, 381, 390

Offset: 0

Hieronymus Fischer, May 19 2007

nonn

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

Other related sequences: A000032, A130241, A130243, A130245, A130246, A130248, A130252, A130258, A130262. Fibonacci inverse see A130233 - A130240, A104162.

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

Join[{0, 0}, Table[Sum[Ceiling[Log[GoldenRatio, k + 1/2]], {k, 0, n}], {n, 1, 50}]] (* G. C. Greubel, Sep 12 2018 *)

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

a(n) = Sum_{k=0..n} A130242(k).
a(n) = n*A130242(n) - A000032(A130242(n) +1) for n>=3.
G.f.: x/(1-x)^2*(2*x^2 + Sum{k>=2, x^Lucas(k)}).

cp's OEIS Frontend

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

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A130248 Partial sums of the Lucas Inverse A130247.

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

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A130235 Partial sums of the 'lower' Fibonacci Inverse A130233.

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

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A130246 Partial sums of A130245.

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A130242 Minimal index k of a Lucas number such that Lucas(k)>=n (the 'upper' Lucas (A000032) Inverse).

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