A226857 -id:A226857 - 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

A059389 Sums of two nonzero Fibonacci numbers.

Original entry on oeis.org

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, 254, 267

Offset: 1

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

The sums of two distinct nonzero Fibonacci numbers is essentially the same sequence: 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 18, 21, ... (only 2 is missing), since F(i) + F(i) = F(i-2) + F(i+1). - Colm Mulcahy, Mar 02 2008

To elaborate on Mulcahy's comment above: all terms of A078642 are in this sequence; those are numbers with two distinct representations as the sum of two Fibonacci numbers, which are, as Alekseyev proved, numbers of the form 2*F(i) greater than 2. - Alonso del Arte, Jul 07 2013

			10 is in the sequence because 10 = 2 + 8.
11 is in the sequence because 11 = 3 + 8.
12 is not in the sequence because no pair of Fibonacci numbers adds up to 12.

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

Cf. A000045, A059390 (complement). Similar in nature to A048645. Essentially the same as A084176. Intersection with A049997 is A226857.

N:= 1000: # to get all terms <= N
R:= NULL:
for j from 1 do
  r:= combinat:-fibonacci(j);
  if r > N then break fi;
  R:= R, r;
end:
R:= {R}:
select(`<=`, {seq(seq(r+s, s=R),r=R)},N);
# if using Maple 11 or earlier, uncomment the next line
# sort(convert(%,list)); # Robert Israel, Feb 15 2015

max = 13; Select[Union[Total/@Tuples[Fibonacci[Range[2, max]], {2}]], # <= Fibonacci[max] &] (* Harvey P. Dale, Mar 13 2011 *)

list(lim)=my(upper=log(lim*sqrt(5))\log((1+sqrt(5))/2)+1, t, tt, v=List([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

a(1) = 2 and for n >= 2 a(n) = F_(trinv(n-2)+2) + F_(n-((trinv(n-2)*(trinv(n-2)-1))/2)) where F_n is the n-th Fibonacci number, F_1 = 1 F_2 = 1 F_3 = 2 ... and the definition of trinv(n) is in A002262. - Noam Katz (noamkj(AT)hotmail.com), Feb 04 2001

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

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

A049999 a(n) = smallest index k such that Fibonacci(k) = d(n), where d = A049998 (sequence of first differences of ordered products of Fibonacci numbers, i.e., of A049997, with no duplicates).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 1, 1, 4, 3, 1, 5, 4, 1, 1, 6, 5, 1, 3, 7, 6, 1, 1, 4, 8, 7, 3, 1, 5, 9, 8, 4, 1, 1, 6, 10, 9, 5, 1, 3, 7, 11, 10, 6, 1, 1, 4, 8, 12, 11, 7, 3, 1, 5, 9, 13, 12, 8, 4, 1, 1, 6, 10, 14, 13, 9, 5, 1, 3, 7, 11, 15, 14, 10, 6, 1, 1, 4, 8, 12, 16, 15, 11

Offset: 1

Clark Kimberling

nonn

"David W. Wilson conjectured (Dec 14 2005) that" sequence A049998 "consists only of Fibonacci numbers. Proofs were found by Franklin T. Adams-Watters and Don Reble, Dec 14 2005." - Petros Hadjicostas, Nov 08 2019 [This comment was copied from A049998, which includes Don Reble's proof of the conjecture.]

			From _Petros Hadjicostas_, Nov 08 2019: (Start)
A049998(1) = 1 = Fibonacci(1) = Fibonacci(2), so a(1) = min(1,2) = 1.
A049998(7) = 2 = Fibonacci(3), so a(7) = 3.
A049998(10) = 3 = Fibonacci(4), so a(10) = 4.
A049998(13) = 5 = Fibonacci(5), so a(13) = 5.
A049998(17) = 8 = Fibonacci(6), so a(17) = 6. (End)

Clark Kimberling, Orderings of products of Fibonacci numbers, Fibonacci Quarterly 42:1 (2004), pp. 28-35. (Includes a proof of the conjecture proved in the Comments for sequence A049998.)

Cf. A000045, A049997, A049998, A094563, A226857, A271354.

A000045(a(n)) = A049998(n) = A049997(n) - A049997(n-1) for n >= 1. - Petros Hadjicostas, Nov 08 2019

Name edited by and more terms from Petros Hadjicostas, Nov 08 2019

cp's OEIS Frontend

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

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A059389 Sums of two nonzero Fibonacci numbers.

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A049999 a(n) = smallest index k such that Fibonacci(k) = d(n), where d = A049998 (sequence of first differences of ordered products of Fibonacci numbers, i.e., of A049997, with no duplicates).

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