A084176 -id:A084176 - cp's OEIS frontend

A059389 Sums of two nonzero Fibonacci numbers.

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

A280154 a(n) = 5*Lucas(n).

Original entry on oeis.org

10, 5, 15, 20, 35, 55, 90, 145, 235, 380, 615, 995, 1610, 2605, 4215, 6820, 11035, 17855, 28890, 46745, 75635, 122380, 198015, 320395, 518410, 838805, 1357215, 2196020, 3553235, 5749255, 9302490, 15051745, 24354235, 39405980, 63760215, 103166195, 166926410, 270092605, 437019015

Offset: 0

Bruno Berselli, Dec 27 2016

Fibonacci sequence beginning 10, 5.
After 5, the sequence provides the 3rd column of the rectangular array in A213590.
After 5, all terms belong to A191921 because a(n) = Lucas(n+4) - 3*Lucas(n-1).
From G. C. Greubel, Dec 27 2016: (Start)
{a(n) mod 3} yields (1,2,0,2,2,1,0,1), repeated, and is given as A082115.
{a(n) mod 6} yields (4,5,3,2,5,1,0,1,1,2,3,5,2,1,3,4,1,5,0,5,5,4,3,1) and is given as A082117. (End)

Bruno Berselli, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (1,1).

Subsequence of A084176.
Cf. A022088: 5*Fibonacci(n).
Cf. A022359: Lucas(n+5) + Lucas(n-5).
Cf. A000032, A000045, A191921, A213590.
Cf. sequences with formula Fibonacci(n+k) + Fibonacci(n-k): A006355 (k=0, without the initial 1), A000032 (k=1), A022086 (k=2), A022112 (k=3, with an initial 4), A022090 (k=4), this sequence (k=5), A022352 (k=6).

Magma
```
[5*Lucas(n): n in [0..40]];
```

F := n -> combinat:-fibonacci(n):
seq(F(n+5) + F(n-5), n=0..38); # Peter Luschny, Dec 29 2016

Mathematica
```
Table[5 LucasL[n], {n, 0, 40}]
```

vector(40, n, n--; fibonacci(n+5)+fibonacci(n-5))

def A280154():
    x, y = 10, 5
    while True:
        yield x
        x, y = y, x + y
a = A280154(); print([next(a) for  in range(39)]) # _Peter Luschny, Dec 29 2016

G.f.: 5*(2 - x)/(1 - x - x^2).
a(n) = a(n-1) + a(n-2) for n>1.
a(n) = Fibonacci(n+5) + Fibonacci(n-5), with Fibonacci(-k) = -(-1)^k*Fibonacci(k) for the negative indices.

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

A272575 Perfect powers that are the sum of two Fibonacci numbers.

Original entry on oeis.org

1, 4, 8, 9, 16, 36, 144, 1000, 1600, 14930496

Offset: 1

Altug Alkan, May 03 2016

nonn

Intersection of A001597 and A084176.
Listed terms are 1, 2^2, 2^3, 3^2, 2^4, 6^2, 12^2, 10^3, 40^2, 3864^2.
First five terms are also members of A000961.
Conjecture: there are no more terms in this sequence. Any remaining terms must have over 10000 digits. - Charles R Greathouse IV, May 04 2016

			8 is a term because 2^3 = 3 + 5.

Cf. A000045, A001597, A084176, A111378.

Select[Range[10^4], Function[k, Or[k == 1, GCD @@ Map[Last, FactorInteger@ k] > 1] && Total@ Map[Times @@ Boole@ Map[MemberQ[s, #] &, #] &, Transpose@ {#, k - #} &@ Range[0, Floor[k/2]]] > 0]] (* Michael De Vlieger, May 03 2016 *)

list(lim)=my(upper=log(lim*sqrt(5))\log((1+sqrt(5))/2)+1, t, tt, v=List([1])); 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); if(ispower(tt), listput(v, tt)))); Set(v) \\ Charles R Greathouse IV, May 03 2016

A272632 Non-Fibonacci numbers that are both a sum and a difference of two Fibonacci numbers.

Original entry on oeis.org

4, 6, 7, 10, 11, 16, 18, 26, 29, 42, 47, 68, 76, 110, 123, 178, 199, 288, 322, 466, 521, 754, 843, 1220, 1364, 1974, 2207, 3194, 3571, 5168, 5778, 8362, 9349, 13530, 15127, 21892, 24476, 35422, 39603, 57314, 64079, 92736, 103682, 150050, 167761, 242786

Offset: 1

Altug Alkan, May 04 2016

Intersection of A001690 and A007298 and A084176.
Sequence focuses on the non-Fibonacci numbers because of the fact that all Fibonacci numbers are both the sum of two Fibonacci numbers and the difference of two Fibonacci numbers by definition of Fibonacci numbers.
For relation with Lucas numbers, see formula section.

			6 is a term because 6 = Fibonacci(1) + Fibonacci(5) = Fibonacci(6) - Fibonacci(3).
16 is a term because 16 = Fibonacci(6) + Fibonacci(6) = Fibonacci(8) - Fibonacci(5).
167761 is a term because it is not a Fibonacci number and 167761 = Fibonacci(24) + Fibonacci(26) = 46368 + 121393 and Fibonacci(24) + Fibonacci(26) = Fibonacci(27) - Fibonacci(23) by definition.

Index entries for linear recurrences with constant coefficients, signature (0,1,0,1).

Cf. A000032, A055389, A001690, A007298, A084176.

mxf=30; {s,d} = Reap[Do[{a,b} = Fibonacci@{i,j}; Sow[a+b, 0]; Sow[a-b, 1], {i, mxf}, {j, i}]][[2]]; Complement[ Intersection[s, d], Fibonacci@ Range@ mxf] (* Giovanni Resta, May 04 2016 *)

a(2*n-1) = fibonacci(n+1) + fibonacci(n+3) =A000204(n+2) for n >= 1.
a(2*n) = 2*fibonacci(n+3)  = A078642(n+1) for n >= 1.
G.f.: -x*(4+6*x+3*x^2+4*x^3)/(-1+x^2+x^4) . - R. J. Mathar, Jan 13 2023
a(n) = a(n-2) + a(n-4) for n > 4. - Christian Krause, Oct 31 2023

A272635 Numbers that are not a sum or a difference of two Fibonacci numbers.

Original entry on oeis.org

17, 25, 27, 28, 30, 38, 40, 41, 43, 44, 45, 46, 48, 49, 51, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 77, 78, 79, 80, 82, 83, 85, 93, 95, 96, 98, 99, 100, 101, 103, 104, 105, 106, 107, 108, 109, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122

Offset: 1

Altug Alkan, May 04 2016

This sequence is the complement of the union of A007298 and A084176.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000045, A007298, A084176, A105448, A105449.

N:= 30: # to get all terms < A000045(N+1) - A000045(N-4)
fibs:= [seq(combinat:-fibonacci(i),i=1..N)]:
R:= {seq(seq(fibs[i]-fibs[j],j=1..i-1),i=1..N), seq(seq(fibs[i]+fibs[j],j=1..i),i=1..N)}:
A:= {$1..fibs[-1]+fibs[-2]-fibs[-5]-1} minus R:
sort(convert(A,list)); # Robert Israel, May 04 2016

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

A377536 Integers that are the arithmetic mean of two distinct Fibonacci numbers (A000045).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 17, 18, 21, 28, 29, 30, 34, 38, 45, 46, 47, 51, 55, 72, 73, 76, 89, 117, 118, 119, 123, 127, 144, 161, 189, 190, 191, 195, 199, 216, 233, 305, 306, 309, 322, 377, 494, 495, 496, 500, 504, 521, 538, 610, 682, 799, 800, 801, 805

Offset: 1

Felix Huber, Dec 18 2024

nonn

This sequence contains all positive Fibonacci numbers of A000045. Proof: For i >= 2, (F(i-2) + F(i+1))/2 = (F(i-2) + F(i-1) + F(i))/2 = (F(i-2) + F(i-1) + F(i-2) + F(i-1))/2 = F(i-1) + F(i-2) = F(i).

			1 is in the sequence because (F(0) + F(3))/2 = (0 + 2)/2 = 1.
12 is in the sequence because (F(4) + F(8))/2 = (3 + 21)/2 = 12.

Felix Huber, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Fibonacci Number

Cf. A000045, A084176.

with(combinat):
A377536:=proc(k)
   local L,M,i,j;
   M:={};
   L:=[seq(fibonacci(i),i=0..k)];
   for i to k do
      for j from i+1 to k+1 do
         if is(L[i]+L[j],even) then
            M:=[op(M),(L[i]+L[j])/2]
         fi
      od
   od;
   M:=convert(M,set);
   return op(M)
end proc:
A377536(17)

A272589 Numbers n such that the equation F(n) = sigma(F(i) + F(j)) has a solution with i >= 1 and j >= 0, where F(k) = A000045(k) represents the k-th Fibonacci number.

Original entry on oeis.org

1, 2, 4, 6, 7, 12, 24

Offset: 1

Altug Alkan, May 03 2016

Corresponding distinct F(n) values for listed terms are 1, 3, 8, 13, 144, 46368.
Corresponding F(i) + F(j) values are for listed terms are 1, 2, 7, 9, 94, 28678.
It is known that for almost all positive integers n, the sum of divisors of Fibonacci(n) is not a Fibonacci number (see A272412). This sequence focuses on the sums of two Fibonacci numbers for a similar question. Since A000045 is obvious subsequence of A084176 by definition of Fibonacci numbers, the reason of existence of this sequence can be seen as a generalized version of question that is motivation of A272412.

			7 is a term because Fibonacci(7) = 13 = sigma(1 + 8) and 1, 8 are Fibonacci numbers.

Cf. A000203, A000045, A084176, A272412.

A344661 Integers k such that k^2 is the sum of two Fibonacci numbers.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 12, 40, 3864

Offset: 1

Lamine Ngom, May 26 2021

Is this sequence finite?

No other terms below 10^20899.

			These square sums of Fibonacci numbers correspond to:
     0^2 = F(0)  + F(0);
     1^2 = F(1)  + F(0)  = F(2) + F(0);
     2^2 = F(4)  + F(1)  = F(4) + F(2) = F(3) + F(3);
     3^2 = F(6)  + F(1)  = F(6) + F(2);
     4^2 = F(7)  + F(4)  = F(6) + F(6);
     6^2 = F(9)  + F(3);
    12^2 = F(11) + F(10) = F(12) + F(0);
    40^2 = F(17) + F(4);
  3864^2 = F(36) + F(12).

F. Luca and V. Patel, On perfect powers that are sums of two Fibonacci numbers, J. Number Theory, 189:90-96, 2018.

Cf. A000045, A007298, A084176, A111378, A219114.

cp's OEIS Frontend

A059389 Sums of two nonzero Fibonacci numbers.

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A280154 a(n) = 5*Lucas(n).

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

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A272575 Perfect powers that are the sum of two Fibonacci numbers.

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A272632 Non-Fibonacci numbers that are both a sum and a difference of two Fibonacci numbers.

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A272635 Numbers that are not a sum or a difference of two Fibonacci numbers.

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Maple

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

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Maple