A059389 -id:A059389 - cp's OEIS frontend

A049997 Numbers of the form Fibonacci(i)*Fibonacci(j).

0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 13, 15, 16, 21, 24, 25, 26, 34, 39, 40, 42, 55, 63, 64, 65, 68, 89, 102, 104, 105, 110, 144, 165, 168, 169, 170, 178, 233, 267, 272, 273, 275, 288, 377, 432, 440, 441, 442, 445, 466, 610, 699, 712, 714, 715, 720, 754

Offset: 0

Clark Kimberling

It follows from Atanassov et al. that a(n) << sqrt(phi)^n, which matches the trivial a(n) >> sqrt(phi)^n up to a constant factor. - Charles R Greathouse IV, Feb 06 2013

Conjecture: Fibonacci(m)*Fibonacci(n) with 2 < m < n is a perfect power only for (m,n) = (3,6). This has been verified for 2 < m < n <= 900. - Zhi-Wei Sun, Jan 02 2025

			25 is in the sequence since it is the product of two, not necessarily distinct, Fibonacci numbers, 5 and 5.
26 is in the sequence since it is the product of two Fibonacci numbers, 2 and 13.
27 is not in the sequence because there is no way whatsoever to represent it as the product of exactly two Fibonacci numbers.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
K. T. Atanassov, Ron Knott, Kiyota Ozeki, A. G. Shannon, and László Szalay, Inequalities among related pairs of Fibonacci numbers, Fibonacci Quarterly 41:1 (2003), pp. 20-22.
Clark Kimberling, Orderings of products of Fibonacci numbers, Fibonacci Quarterly 42:1 (2004), pp. 28-35.
Zhi-Wei Sun, Perfect powers as products of two Fibonacci or Lucas numbersQuestion 485037 at MathOverflow, December 30, 2024.

Subsequence of A065108; apart from the first term, subsequence of A094563. Complement is A228523.
See A049998 for further information about this sequence. Cf. A080097.
Intersection with A059389 (sums of two Fibonacci numbers) is A226857.
Cf. also A090206, A005478.

Take[ Union@Flatten@Table[ Fibonacci[i]Fibonacci[j], {i, 0, 16}, {j, 0, i}], 61] (* Robert G. Wilson v, Dec 14 2005 *)

list(lim)=my(phi=(1+sqrt(5))/2, v=vector(log(lim*sqrt(5))\log(phi), i, fibonacci(i+1)), u=List([0]),t); for(i=1,#v,for(j=i,#v,t=v[i]*v[j];if(t>lim,break,listput(u,t)))); vecsort(Vec(u),,8) \\ Charles R Greathouse IV, Feb 05 2013

A084176 Sums of two Fibonacci numbers (allowing 0 as a summand).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 18, 21, 22, 23, 24, 26, 29, 34, 35, 36, 37, 39, 42, 47, 55, 56, 57, 58, 60, 63, 68, 76, 89, 90, 91, 92, 94, 97, 102, 110, 123, 144, 145, 146, 147, 149, 152, 157, 165, 178, 199, 233, 234, 235, 236, 238, 241, 246

Offset: 1

Jon Perry, Jun 20 2003

			14=13+1, so is in, however 17 is not expressible as Fib(i)+Fib(j).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000045. Essentially the same as A059389.

list(lim)=my(upper=log(lim*sqrt(5))\log((1+sqrt(5))/2)+1, t, tt, v=List([0,1,2])); if(fibonacci(t)>lim,t--); for(i=3,upper, t=fibonacci(i); for(j=2,i-1,tt=t+fibonacci(j); if(tt>lim, break, listput(v,tt)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 24 2012

log a(n) ~ sqrt(n log phi) where phi is the golden ratio A001622. There are (log x/log phi)^2 + O(log x) members of this sequence up to x. - Charles R Greathouse IV, Jul 24 2012

A111378 Squares that are equal to the sum of two Fibonacci numbers.

Original entry on oeis.org

0, 1, 4, 9, 16, 36, 144, 1600, 14930496

Offset: 1

Giovanni Teofilatto, Nov 09 2005

nonn

Any further terms have more than 10,000 digits. - Charles R Greathouse IV, Sep 16 2015

Squares in A084176 (or A059389). Cf. A000045, A179334.

Fibs:= {seq(combinat:-fibonacci(i),i=0..100)}:
sort(convert(select(issqr,{seq(seq(Fibs[i]+Fibs[j],j=1..i),i=1..100)}),list)); # Robert Israel, Jun 03 2024

Select[Union[Total/@Subsets[Fibonacci[Range[0,100]],{2}],Table[Fibonacci[n]*2,{n,0,100}]],IntegerQ[Sqrt[#]]&] (* James C. McMahon, Jun 03 2024 *)

list(lim)=my(F=List(),v=List([0,1]),n=1,t); while((t=fibonacci(n++))<=lim, listput(F,t)); F=Vec(F); for(i=1,#F,for(j=i,#F, if(issquare(t=F[i]+F[j]), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Sep 16 2015

1600 from Jonathan Vos Post, Nov 11 2005

14930496 from N. J. A. Sloane, Nov 11 2005

A178576 Primes that are the sum of two Fibonacci numbers.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 23, 29, 37, 47, 89, 97, 149, 157, 199, 233, 241, 379, 521, 613, 631, 1021, 1597, 1741, 2207, 3571, 9349, 10949, 11933, 17713, 28657, 46381, 46457, 46601, 50549, 75169, 196439, 203183, 214129, 514229, 560597, 832129, 2178343

Offset: 1

Carmine Suriano, Jan 12 2011

nonn

The corresponding prime indices are in A178971.

			Prime 613 can be expressed as 3+610 = Fibonacci(4)+Fibonacci(15), therefore 613 is in the sequence.

Charles R Greathouse IV, Table of n, a(n) for n = 1..5624

Cf. A000040, A000045, A178971. Subsequence of A059389.

f=Fibonacci[Range[33]]; Select[Union[Flatten[Outer[Plus, f, f]]], PrimeQ]

list(lim)={
    my(v,u=List(),t);
    v=vector(log(lim*sqrt(5))\log((1+sqrt(5))/2)+1,n,fibonacci(n));
    for(i=1,#v,for(j=i+1,#v,
        t = v[i] + v[j];
        if(t > lim, break);
        if(ispseudoprime(t),listput(u, t))
    ));
    vecsort(Vec(u),,8)
}; \\ Charles R Greathouse IV, Jul 23 2012

A226857 Numbers that are both the sum of two Fibonacci numbers and the product of two Fibonacci numbers.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 13, 15, 16, 21, 24, 26, 34, 39, 42, 55, 63, 68, 89, 102, 110, 144, 165, 178, 233, 267, 288, 377, 432, 466, 610, 699, 754, 987, 1131, 1220, 1597, 1830, 1974, 2584, 2961, 3194, 4181, 4791, 5168, 6765, 7752, 8362, 10946, 12543, 13530

Offset: 1

Alonso del Arte, Jun 19 2013

All Fibonacci numbers are in the sequence. The only prime numbers in this sequence are prime Fibonacci numbers.

			5 + 21 = 2 * 13 = 26, therefore 26 is in the sequence.
8 + 21 = 1 * 34 = 34, therefore 34 is in the sequence.
5 + 34 = 3 * 13 = 39, therefore 39 is in the sequence.

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

Cf. A059389, A049997.

t = Fibonacci[Range[0, 25]]; t1 = Select[Union[Flatten[Table[a + b, {a, t}, {b, t}]]], # <= t[[-1]] &]; t2 = Select[Union[Flatten[Table[a*b, {a, t}, {b, t}]]], # <= t[[-1]] &]; Intersection[t1, t2] (* T. D. Noe, Jul 03 2013 *)

Conjecture: a(n) = a(n-3)+a(n-6) for n>12. - Colin Barker, Nov 09 2014

Empirical g.f.: -x^2*(x^10 +x^9 +x^8 +2*x^7 +3*x^6 +3*x^5 +3*x^4 +3*x^3 +3*x^2 +2*x +1) / (x^6 +x^3 -1). - Colin Barker, Nov 09 2014

A059390 Numbers that are not the sum of two nonzero Fibonacci numbers.

Original entry on oeis.org

1, 12, 17, 19, 20, 25, 27, 28, 30, 31, 32, 33, 38, 40, 41, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 93, 95, 96, 98, 99, 100, 101, 103, 104, 105, 106, 107, 108, 109

Offset: 1

Avi Peretz (njk(AT)netvision.net.il), Jan 29 2001

Except for the initial 1, each term can be expressed as the sum of a Fibonacci number and a non-Fibonacci number (A001690) and is thus the sum of three or more Fibonacci numbers. - Alonso del Arte, Jun 18 2013

That is, aside from the first term, numbers n such that A007895(n) > 2. - Charles R Greathouse IV, Jun 18 2013

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Complement of A059389.

Complement[ Range[ Fibonacci[ 12 ]], Union[ Flatten[ Table[ Fibonacci[i] + Fibonacci[j], {i, 12}, {j, i - 1}]]]] (* Robert G. Wilson v, Jul 22 2005 *)

is(n)=if(n<9,return(n==1)); my(k);while(fibonacci(k++)<=n,); n-=fibonacci(k-1); k=n^2; k+=(k+1)<<2; !issquare(k) && !issquare(k-8) \\ Charles R Greathouse IV, Jun 18 2013

More terms from Larry Reeves (larryr(AT)acm.org), Jan 31 2001

A362295 Sums of two Fibonacci numbers that are also sums of two squares.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 9, 10, 13, 16, 18, 26, 29, 34, 36, 37, 58, 68, 89, 90, 97, 144, 145, 146, 149, 157, 178, 233, 234, 241, 288, 377, 466, 521, 610, 612, 613, 754, 1000, 1021, 1042, 1076, 1220, 1597, 1600, 1602, 1618, 1741, 2592, 2597, 2605, 2817, 3194, 4181, 4194, 4325, 6770, 6773, 6778, 6786

Offset: 1

Robert Israel, Apr 14 2023

nonn

Intersection of A001481 and A084176.

			a(5) = 5 is a term because 5 = 2 + 3 = A000045(3) + A000045(4) = 2^2 + 1^2.

Robert Israel, Table of n, a(n) for n = 1..2500

Cf. A000045, A001481, A059389, A084176, A111378.

ss:= proc(n) local F,t;
  F:= ifactors(n)[2];
  andmap(t ->  t[1] mod 4 <> 3 or t[2]::even, F)
end proc:
fibs:= map(combinat:-fibonacci, {$0..25}):
N:= max(fibs):
fib2:= {seq(seq(fibs[i]+fibs[j],i=1..j),j=1..nops(fibs))}:
sort(convert(select(t -> t <= N and ss(t), fib2),list));

max = 150; (* max = 150 gives 1670 terms *)
Join[{0, 1}, Select[Union[Total /@ Tuples[Fibonacci[Range[2, max]], {2}]], # <= Fibonacci[max] && SquaresR[2, #] != 0&]] (* Jean-François Alcover, Sep 29 2024, after Harvey P. Dale in A059389 *)

from itertools import islice
from sympy import factorint
def A362295_gen(): # generator of terms
    yield from (0,1,2)
    a = [1,2]
    while True:
        b = a[-1]+a[-2]
        c = a[-1]<<1
        flag = True
        for d in a:
            n = b+d
            if flag and n>=c:
                if n>c:
                   f = factorint(c)
                   if all(d & 3 != 3 or f[d] & 1 == 0 for d in f):
                       yield c
                flag = False
            f = factorint(n)
            if all(d & 3 != 3 or f[d] & 1 == 0 for d in f):
                yield n
        a.append(b)
A362295_list = list(islice(A362295_gen(),60)) # Chai Wah Wu, Apr 16 2023

cp's OEIS Frontend

A049997 Numbers of the form Fibonacci(i)*Fibonacci(j).

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A084176 Sums of two Fibonacci numbers (allowing 0 as a summand).

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A111378 Squares that are equal to the sum of two Fibonacci numbers.

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A178576 Primes that are the sum of two Fibonacci numbers.

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A226857 Numbers that are both the sum of two Fibonacci numbers and the product of two Fibonacci numbers.

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A059390 Numbers that are not the sum of two nonzero Fibonacci numbers.

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A362295 Sums of two Fibonacci numbers that are also sums of two squares.

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