A001690 Non-Fibonacci numbers. (Formerly M3268 N1319)

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

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

T. D. Noe, Table of n, a(n) for n = 1..1000

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

Adjacent sequences: A001687 A001688 A001689 * A001691 A001692 A001693

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Status

approved