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A001690 Non-Fibonacci numbers.
(Formerly M3268 N1319)
33
4, 6, 7, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
A010056(a(n)) = 0. - Reinhard Zumkeller, Oct 10 2013
REFERENCES
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Bakir Farhi, An explicit formula generating the non-Fibonacci numbers, arXiv:1105.1127 [math.NT], May 05 2011.
H. W. Gould, Non-Fibonacci numbers, Fib. Quart., 3 (1965), pp. 177-183.
FORMULA
a(n-1) = floor(n + lgg(sqrt(5)*(lgg(sqrt(5)*n)+n) - 5 + 3/n) - 2) where lgg(x) = log(x)/log((sqrt(5)+1)/2), given by Farhi. - Jonathan Vos Post, May 05 2011
a(n) ~ n. - Charles R Greathouse IV, Nov 06 2014
a(n) = floor(1/2 - LambertW(-1, -log(phi)/(sqrt(5)*phi^(n - 3/2)))/log(phi)) with phi = (1 + sqrt(5))/2 [Nicolas Normand (Nantes)]. - Simon Plouffe, Nov 29 2017 [abs removed by Peter Luschny, Nov 30 2017]
MAPLE
a:=proc(n) floor(-LambertW(-1, -1/5*ln(1/2+1/2*5^(1/2))*5^(1/2) /((1/2+1/2*5^(1/2))^(n-3/2))) /ln(1/2+1/2*5^(1/2))+1/2) end:
seq(a(n), n=1..69); # Simon Plouffe, Nov 29 2017
# alternative
isA000045 := proc(n)
for k from 0 do
if A000045(k) = n then
return true;
elif A000045(k) > n then
return false;
end if;
end do:
end proc:
A001690 := proc(n)
option remember;
if n = 1 then
4 ;
else
for a from procname(n-1)+1 do
if not isA000045(a) then
return a;
end if;
end do:
end if;
end proc:
seq(A001690(n), n=1..100) ; # R. J. Mathar, Feb 01 2019
# third Maple program:
q:= n-> (t-> issqr(t+4) or issqr(t-4))(5*n^2):
remove(q, [$1..100])[]; # Alois P. Heinz, Jun 05 2019
MATHEMATICA
Complement[Range[Fibonacci[a = 12]], Fibonacci[Range[a]]] (* Vladimir Joseph Stephan Orlovsky, Jul 01 2011 *)
a[n_] := With[{phi = (1 + Sqrt[5])/2}, Floor[1/2 - LambertW[-1, -Log[phi]/(Sqrt[5] phi^(n - 3/2))]/Log[phi]]];
Table [a[n], {n, 1, 70}] (* Peter Luschny, Nov 30 2017 *)
Table[Floor[n +Log[GoldenRatio, Sqrt[5]*(Log[GoldenRatio, Sqrt[5]*n] +n) -5 +3/n] -2], {n, 2, 100}] (* G. C. Greubel, May 26 2019 *)
PROG
(PARI) lgg(x)=log(x)/log((sqrt(5)+1)/2);
a(n)=n++; floor(n+lgg(sqrt(5)*(lgg(sqrt(5)*n)+n)-5+3/n)-2);
vector(66, n, a(n)) /* Joerg Arndt, May 14 2011 */
(PARI) lower=3; upper=5; for(i=4, 20, for(n=lower+1, upper-1, print1(n", ")); [lower, upper]=[upper, lower+upper]) \\ Charles R Greathouse IV, Nov 19 2013
(Haskell)
a001690 n = a001690_list !! (n-1)
a001690_list = filter ((== 0) . a010056) [0..]
-- Reinhard Zumkeller, Oct 10 2013
(Python)
def f(n):
a=1
b=2
c=3
while n>0:
a=b
b=c
c=a+b
n-=(c-b-1)
n+=(c-b-1)
return (b+n)
for i in range(1, 1001):
print(str(i)+" "+str(f(i))) # Indranil Ghosh, Dec 22 2016
(Magma) phi:= (1+Sqrt(5))/2; [Floor(n + Log(phi, Sqrt(5)*(Log(phi, Sqrt(5)*n) + n) - 5 + 3/n) - 2 ): n in [2..100]]; // G. C. Greubel, May 26 2019
(Sage) [floor( n + log( sqrt(5)*(log(sqrt(5)*n, golden_ratio) + n) - 5 + 3/n , golden_ratio) - 2 ) for n in (2..100)] # G. C. Greubel, May 26 2019
CROSSREFS
The nonnegative integers that are not in A000045.
Cf. A010056.
Sequence in context: A213627 A225871 A288383 * A105447 A242286 A144222
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved

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Last modified July 23 16:21 EDT 2024. Contains 374552 sequences. (Running on oeis4.)