A130242 -id:A130242 - cp's OEIS frontend

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

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)}).

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

Original entry on oeis.org

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).

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

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).

A130252 Partial sums of A130250.

Original entry on oeis.org

0, 1, 4, 7, 11, 15, 20, 25, 30, 35, 40, 45, 51, 57, 63, 69, 75, 81, 87, 93, 99, 105, 112, 119, 126, 133, 140, 147, 154, 161, 168, 175, 182, 189, 196, 203, 210, 217, 224, 231, 238, 245, 252, 259, 267, 275, 283, 291, 299, 307, 315, 323, 331, 339, 347, 355, 363, 371

Offset: 0

Hieronymus Fischer, May 20 2007

nonn

If the initial zero is omitted, partial sums of A130253.

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

Cf. A130249, A130251, A130253, A130252, A130234, A130236, A130242, A130244.

Cf. A001045, A105348.

A001045:= func< n | (2^n - (-1)^n)/3 >;
A130252:= func< n | n eq 0 select 0 else (2*n*Ceiling(Log(2, 3*n-1)) - A001045(Ceiling(Log(2,3*n-1)) +1) +1)/2 >;
[A130252(n): n in [0..70]]; // G. C. Greubel, Mar 18 2023

A001045[n_]:= (2^n - (-1)^n)/3;
A130252[n_]:= If[n==0, 0, (2*n*Ceiling[Log[2,3*n-1]] - A001045[Ceiling[Log[2,3*n-1]]+1] +1)/2];
Table[A130252[n], {n,0,70}] (* G. C. Greubel, Mar 18 2023 *)

def A130252(n): return n*(m:=(3*n-1).bit_length())-(((1<>1) # Chai Wah Wu, Apr 17 2025

def A001045(n): return (2^n - (-1)^n)/3
def A130252(n): return 0 if (n==0) else (2*n*ceil(log(3*n-1,2)) - A001045(ceil(log(3*n-1,2)) +1) +1)/2
[A130252(n) for n in range(71)] # G. C. Greubel, Mar 18 2023

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

A130250 Minimal index k of a Jacobsthal number such that A001045(k) >= n (the 'upper' Jacobsthal inverse).

Original entry on oeis.org

0, 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

nonn

Inverse of the Jacobsthal sequence (A001045), nearly, since a(A001045(n))=n except for n=2 (see A130249 for another version). a(n+1) is equal to the partial sum of the Jacobsthal indicator sequence (see A105348).

			a(10)=5 because A001045(5) = 11 >= 10, but A001045(4) = 5 < 10.

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

For partial sums see A130252.

Cf. A001045, A105348, A130234, A130242, A130249, A130253.

[0] cat [Ceiling(Log(2,3*n-1)): n in [1..120]]; // G. C. Greubel, Mar 18 2023

Table[If[n==0, 0, Ceiling[Log[2, 3*n-1]]], {n,0,120}] (* G. C. Greubel, Mar 18 2023 *)

def A130250(n): return (3*n-2).bit_length() if n else 0 # Chai Wah Wu, Apr 17 2025

def A130250(n): return 0 if (n==0) else ceil(log(3*n-1, 2))
[A130250(n) for n in range(121)] # G. C. Greubel, Mar 18 2023

a(n) = ceiling(log_2(3n-1)) = 1 + floor(log_2(3n-2)) for n >= 1.
a(n) = A130249(n-1) + 1 = A130253(n-1) for n >= 1.
G.f.: (x/(1-x))*Sum_{k>=0} x^A001045(k).

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.

Original entry on oeis.org

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.

cp's OEIS Frontend

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

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

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A130252 Partial sums of A130250.

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