A001690 Non-Fibonacci numbers.

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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.

Crossrefs

The nonnegative integers that are not in A000045.

Cf. A010056.

Programs

Haskell

a001690 n = a001690_list !! (n-1)
a001690_list = filter ((== 0) . a010056) [0..]
-- Reinhard Zumkeller, Oct 10 2013

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

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 *)

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

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

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

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]