A130251 -id:A130251 - 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).

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

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]

A130240 Partial sums of A130239.

Original entry on oeis.org

0, 2, 4, 6, 9, 12, 15, 18, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61, 65, 69, 73, 77, 81, 85, 90, 95, 100, 105, 110, 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, 255, 260

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, A130236, A130237, A130238, A130239, A130243, A130246, A130248, A130239, A130251, A130253, A130257, A130261.

A130233:= func< n | Floor(Log(3/2 + n*Sqrt(5))/Log((1+Sqrt(5))/2)) >;
A130240:= func< n | (&+[A130233(Floor(Sqrt(j))): j in [0..n]]) >;
[A130240(n): n in [0..70]]; // G. C. Greubel, Mar 18 2023

A130233[n_]:= Floor[Log[GoldenRatio, 3/2 + n*Sqrt[5]]];
A130240[n_]:= A130240[n]= Sum[A130233[Floor[Sqrt[j]]], {j,0,n}];
Table[A130240[n], {n,0,70}] (* G. C. Greubel, Mar 18 2023 *)

def A130233(n): return int(log(3/2 +n*sqrt(5), golden_ratio))
def A130240(n): return sum( A130233(floor(sqrt(j))) for j in range(n+1) )
[A130240(n) for n in range(71)] # G. C. Greubel, Mar 18 2023

a(n) = Sum_{k=0..n} A130239(k).
a(n) = (n+1)*A130233(sqrt(n)) - Fib(A130233(sqrt(n)) + 1) * Fib(A130232(sqrt(n))).
G.f.: (1/(1-x)^2) * Sum_{k>=1} x^(Fib(k)^2).

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).

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).

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).

A130238 Partial sums of A130237.

Original entry on oeis.org

0, 2, 8, 20, 36, 61, 91, 126, 174, 228, 288, 354, 426, 517, 615, 720, 832, 951, 1077, 1210, 1350, 1518, 1694, 1878, 2070, 2270, 2478, 2694, 2918, 3150, 3390, 3638, 3894, 4158, 4464, 4779, 5103, 5436, 5778, 6129, 6489, 6858, 7236, 7623, 8019, 8424, 8838

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, A130236, A130237, A130239, A130240, A130243, A130246, A130248, A130239, A130251, A130253, A130257, A130261.

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

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

def A130238(n): return sum(j*int(log(3/2 +j*sqrt(5), golden_ratio)) for j in range(n+1))
[A130238(n) for n in range(71)] # G. C. Greubel, Mar 18 2023

a(n) = Sum_{k=0..n} A130237(k).
a(n) = (n*(n+1)*A130233(n) - (Fib(A130233(n)) - 1)*(Fib(A130233(n) + 1) - 1))/2.
G.f.: (1/(1-x)^3)*Sum_{k>=1} (Fib(k)*(1-x) + x)*x^Fib(k).

cp's OEIS Frontend

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

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A130240 Partial sums of A130239.

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

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