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Search: a130240 -id:a130240
Displaying 1-10 of 13 results found. page 1 2
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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
A130245 Number of Lucas numbers (A000032) <= n. +10
16
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Partial sums of the Lucas indicator sequence A102460.
For n>=2, we have a(A000032(n)) = n + 1.
LINKS
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.
FORMULA
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.
EXAMPLE
a(9)=5 because there are 5 Lucas numbers <=9 (2,1,3,4 and 7).
MATHEMATICA
Join[{0}, Table[1+Floor[Log[GoldenRatio, (2*n+1)/2]], {n, 1, 100}]] (* G. C. Greubel, Sep 09 2018 *)
PROG
(PARI)
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
(Magma) [0] cat [1+Floor(Log((2*n+1)/2)/Log((1+Sqrt(5))/2)): n in [1..100]]; // G. C. Greubel, Sep 09 2018
(Python)
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
CROSSREFS
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.
KEYWORD
nonn
AUTHOR
Hieronymus Fischer, May 19 2007, Jul 02 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
A130243 Partial sums of the 'lower' Lucas Inverse A130241. +10
11
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..2500
FORMULA
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).
MATHEMATICA
Table[1 + Sum[Floor[Log[GoldenRatio, k + 1/2]], {k, 1, n}], {n, 1, 50}] (* G. C. Greubel, Sep 13 2018 *)
PROG
(PARI) 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
(Magma) [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
CROSSREFS
Other related sequences: A000032, A130244, A130242, A130245, A130246, A130248, A130251, A130257, A130261. Fibonacci inverse see A130233 - A130240, A104162.
KEYWORD
nonn
AUTHOR
Hieronymus Fischer, May 19 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
A130242 Minimal index k of a Lucas number such that Lucas(k)>=n (the 'upper' Lucas (A000032) Inverse). +10
10
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
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).
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..5000
FORMULA
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.
EXAMPLE
a(10)=5, since Lucas(5)=11>=10 but Lucas(4)=7<10.
MATHEMATICA
Join[{0, 0, 0}, Table[Ceiling[Log[GoldenRatio, n + 1/2]], {n, 2, 50}]] (* G. C. Greubel, Dec 24 2017 *)
PROG
(Python)
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
CROSSREFS
For partial sums see A130244.
Other related sequences: A000032, A130241, A130245, A130247, A130250, A130256, A130260.
Indicator sequence A102460.
Fibonacci inverse see A130233 - A130240, A104162.
KEYWORD
nonn
AUTHOR
Hieronymus Fischer, May 19 2007, Jul 02 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
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Last modified August 18 11:30 EDT 2024. Contains 375266 sequences. (Running on oeis4.)