A130243 -id:A130243

A130241

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

+10
25

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

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.

LINKS

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

FORMULA

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.

EXAMPLE

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

MATHEMATICA

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

PROG

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

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

(Python)

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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Hieronymus Fischer, May 19 2007, Jul 02 2007

STATUS

approved

A130248

Partial sums of the Lucas Inverse A130247.

+10
20

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

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

FORMULA

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).

MATHEMATICA

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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Hieronymus Fischer, May 19 2007

STATUS

approved

A130235

Partial sums of the 'lower' Fibonacci Inverse A130233.

+10
14

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

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

FORMULA

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]

MATHEMATICA

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 *)

PROG

(Magma)

m:=120;

f:= func< x | (&+[x^Fibonacci(j): j in [1..Floor(3*Log(3*m+1))]])/(1-x)^2 >;

R<x>:=PowerSeriesRing(Rationals(), m+1);

[0] cat Coefficients(R!( f(x) )); // G. C. Greubel, Mar 17 2023

(SageMath)

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.<x> = PowerSeriesRing(ZZ, prec)

return P( f(x) ).list()

A130235_list(m) # G. C. Greubel, Mar 17 2023

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Hieronymus Fischer, May 17 2007

STATUS

approved

A130240

Partial sums of A130239.

+10
14

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

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

FORMULA

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).

MATHEMATICA

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 *)

PROG

(Magma)

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

(SageMath)

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

CROSSREFS

Cf. A000045, A130233, A130234, A130235, A130236, A130237, A130238, A130239, A130243, A130246, A130248, A130239, A130251, A130253, A130257, A130261.

KEYWORD

nonn

AUTHOR

Hieronymus Fischer, May 17 2007

STATUS

approved

A130246

Partial sums of A130245.

+10
11

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

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

FORMULA

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).

MATHEMATICA

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

PROG

(PARI) 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

(Magma) [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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Hieronymus Fischer, May 19 2007

STATUS

approved

A130237

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

+10
9

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

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

FORMULA

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).

MATHEMATICA

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

PROG

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

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

CROSSREFS

Partial sums: A130238.

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

KEYWORD

nonn

AUTHOR

Hieronymus Fischer, May 17 2007

STATUS

approved

A130239

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

+10
9

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

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

FORMULA

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).

EXAMPLE

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

MATHEMATICA

A130233[n_]:= Floor[Log[GoldenRatio, 3/2 +n*Sqrt[5]]];

Table[A130233[Floor[Sqrt[n]]], {n, 0, 120}] (* G. C. Greubel, Mar 18 2023 *)

PROG

(Magma)

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

(SageMath)

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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Hieronymus Fischer, May 17 2007, May 28 2007

STATUS

approved

A130244

Partial sums of the 'upper' Lucas Inverse A130242.

+10
9

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

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

FORMULA

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)}).

MATHEMATICA

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

PROG

(PARI) 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

(Magma) [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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Hieronymus Fischer, May 19 2007

STATUS

approved

A130238

Partial sums of A130237.

+10
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

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

FORMULA

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).

MATHEMATICA

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 *)

PROG

(Magma) [(&+[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

(SageMath)

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

CROSSREFS

Cf. A000045, A130233, A130234, A130235, A130236, A130237, A130239, A130240, A130243, A130246, A130248, A130239, A130251, A130253, A130257, A130261.

KEYWORD

nonn

AUTHOR

Hieronymus Fischer, May 17 2007

STATUS

approved

A368784

a(0) = 1. For n > 0, a(n) is the smallest integer k > n such that (Sum_{i = 1..n} i)/(Sum_{i = n + 1..k} i) < 1/n.

+10
1

1, 2, 4, 7, 10, 13, 17, 21, 25, 30, 35, 40, 45, 50, 56, 62, 68, 74, 81, 87, 94, 101, 108, 115, 122, 130, 138, 145, 153, 162, 170, 178, 187, 195, 204, 213, 222, 231, 240, 250, 259, 269, 279, 289, 298, 309, 319, 329, 339, 350, 361, 371, 382, 393, 404, 415, 427, 438

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

(Sum_{i = 1..n} i)/(Sum_{i = n + 1..k} i) = n*(n + 1)/((k - n)*(n + 1 + k)) < 1/n. It follows that k > -1/2 + sqrt(4*n^3 + 8*n^2 + 4*n + 1)/2.

LINKS

Felix Huber, Table of n, a(n) for n = 0..10000

FORMULA

a(n) = floor(-1/2 + sqrt(4*n^3 + 8*n^2 + 4*n + 1)/2) + 1.

a(n) = round(sqrt(n*(n+1)^2 + 1/4)). - Chai Wah Wu, Mar 11 2024

EXAMPLE

a(3) = 7, because (1 + 2 + 3)/(4 + 5 + 6 + 7) = 3/11 < 1/3 and (1 + 2 + 3)/(4 + 5 + 6) = 2/5 > 1/3.

MAPLE

A368784 := n -> floor(-1/2 + 1/2*sqrt(4*n^3 + 8*n^2 + 4*n + 1)) + 1;

seq(A368784(n), n = 0 .. 57);

MATHEMATICA

a[n_]:= Floor[-1/2 + Sqrt[4*n^3 + 8*n^2 + 4*n + 1]/2] + 1; Array[a, 58, 0] (* Stefano Spezia, Feb 17 2024 *)

CROSSREFS

Cf. A003067, A061465, A126022, A130243.

KEYWORD

nonn,easy

AUTHOR

Felix Huber, Feb 15 2024

STATUS

approved