A130244 -id:A130244 - cp's OEIS frontend

A130248 Partial sums of the Lucas Inverse A130247.

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).

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]

A130236 Partial sums of the 'upper' Fibonacci Inverse A130234.

Original entry on oeis.org

0, 1, 4, 8, 13, 18, 24, 30, 36, 43, 50, 57, 64, 71, 79, 87, 95, 103, 111, 119, 127, 135, 144, 153, 162, 171, 180, 189, 198, 207, 216, 225, 234, 243, 252, 262, 272, 282, 292, 302, 312, 322, 332, 342, 352, 362, 372, 382, 392, 402, 412, 422, 432, 442, 452, 462, 473

Offset: 0

Hieronymus Fischer, May 17 2007

nonn

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

Cf. A000045, A130233, A130234, A130235, A130244, A130246, A130244, A130246, A130248, A130252, A130258, A130262.

m:=120;
f:= func< x | x*(&+[x^Fibonacci(j): j in [0..Floor(3*Log(3*m+1))]])/(1-x)^2 >;
R:=PowerSeriesRing(Rationals(), m+1);
[0] cat Coefficients(R!( f(x) )); // G. C. Greubel, Mar 18 2023

b[n_]:= For[i=0, True, i++, If[Fibonacci[i] >= n, Return[i]]];
b/@ Range[0, 56]//Accumulate (* Jean-François Alcover, Apr 13 2020 *)

m=120
def f(x): return x*sum( x^fibonacci(j) for j in range(1+int(3*log(3*m+1))))/(1-x)^2
def A130236_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( f(x) ).list()
A130236_list(m) # G. C. Greubel, Mar 18 2023

a(n) = Sum_{k=0..n} A130234(k).
a(n) = n*A130234(n) - Fibonacci(A130234(n)+1) + 1.
G.f.: (x/(1-x)^2) * Sum_{k>=0} x^Fibonacci(k).

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).

A130251 Partial sums of A130249.

Original entry on oeis.org

0, 2, 4, 7, 10, 14, 18, 22, 26, 30, 34, 39, 44, 49, 54, 59, 64, 69, 74, 79, 84, 90, 96, 102, 108, 114, 120, 126, 132, 138, 144, 150, 156, 162, 168, 174, 180, 186, 192, 198, 204, 210, 216, 223, 230, 237, 244, 251, 258, 265, 272, 279, 286, 293, 300, 307, 314, 321

Offset: 0

Hieronymus Fischer, May 20 2007

nonn

			G.f. = 2*x + 4*x^2 + 7*x^3 + 10*x^4 + 14*x^5 + 18*x^6 + 22*x^7 + ... - _Michael Somos_, Sep 17 2018

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

Cf. A130249, A130250, A130252, A130253, A130233, A130235, A130241, A130244. Also A105348, A001045.

[0] cat [(&+[Floor(Log(3*k+1)/Log(2)) : k in [1..n]]): n in [1..100]]; // G. C. Greubel, Sep 09 2018

Join[{0}, Table[Sum[Floor[Log[2, 3*k + 1]], {k, 1, n}], {n, 1, 2500}]] (* G. C. Greubel, Sep 09 2018 *)

for(n=0,100, print1(sum(k=1,n, floor(log(3*k+1)/log(2))), ", ")) \\ G. C. Greubel, Sep 09 2018

def A130251(n): return (n+1)*((m:=(3*n+1).bit_length())-1)-(((1<>1) # Chai Wah Wu, Apr 17 2025

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

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).

cp's OEIS Frontend

A130248 Partial sums of the Lucas Inverse A130247.

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

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A130236 Partial sums of the 'upper' Fibonacci Inverse A130234.

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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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A130251 Partial sums of A130249.

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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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A130252 Partial sums of A130250.