A007298 -id:A007298 - cp's OEIS frontend

A113188 Primes that are the difference of two Fibonacci numbers; primes in A007298.

2, 3, 5, 7, 11, 13, 19, 29, 31, 47, 53, 89, 131, 139, 199, 233, 521, 607, 953, 1453, 1597, 2207, 2351, 2579, 3571, 6763, 9349, 10891, 28513, 28649, 28657, 42187, 44771, 46279, 75017, 189653, 317777, 514229, 1981891, 2177699, 3010349, 3206767

Offset: 1

T. D. Noe, Oct 17 2005

nonn

The difference F(i)-F(j) equals the sum F(j-1)+...+F(i-2) [Corrected by Patrick Capelle, Mar 01 2008]. In general, we need gcd(i,j)=1 for F(i)-F(j) to be prime. The exceptions are handled by the following rule: if i and j are both even or both odd, then F(i)-F(j) is prime if either (1) i-j=4 and L(i-2) is a Lucas prime or (2) i-j=2 and F(i-1) is a Fibonacci prime.

			The prime 139 is here because it is F(12)-F(5).

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

Cf. A000045 (Fibonacci numbers), A001605 (Fibonacci(n) is prime), A001606 (Lucas(n) is prime), A113189 (number of times that Fibonacci(n)-Fibonacci(i) is prime for i=0..n-3).

lst={}; Do[p=Fibonacci[n]-Fibonacci[i]; If[PrimeQ[p], AppendTo[lst, p]], {n, 2, 40}, {i, n-1}]; Union[lst]
Select[Union[Flatten[Differences/@Subsets[Fibonacci[Range[50]],{2}]]],PrimeQ] (* Harvey P. Dale, Aug 04 2024 *)

list(lim)=my(v=List(),F=vector(A130233(lim),i,fibonacci(i)),s,t); for(i=1,#F, s=0; forstep(j=i,1,-1, s+=F[j]; if(s>lim, break); if(isprime(s), listput(v,s)))); Set(v) \\ Charles R Greathouse IV, Oct 07 2016

A219114 Integers n such that n^2 is the difference of two Fibonacci numbers.

Original entry on oeis.org

0, 1, 2, 4, 9, 12, 15, 24

Offset: 1

Max Alekseyev, Nov 12 2012

Numbers n such that n^2 is in A007298.

No other terms below 10^10000. - Manfred Scheucher, Jun 12 2015

			The only known square differences of Fibonacci numbers are:
0^2 = F(2)-F(1) = F(k)-F(k) for any k,
1^2 = F(1)-F(0) = F(2)-F(0) = F(3)-F(1) = F(3)-F(2) = F(4)-F(3),
2^2 = F(5)-F(1) = F(5)-F(2),
4^2 = F(8)-F(5),
9^2 = F(11)-F(6),
12^2 = F(12)-F(0) = F(13)-F(11) = F(14)-F(13),
15^2 = F(13)-F(6),
24^2 = F(15)-F(9).

MathOverflow, Can the difference of two distinct Fibonacci numbers be a square infinitely often?
Manfred Scheucher, Sage Script

Cf. A000045 (Fibonacci numbers).

Cf. A007298 (differences of Fibonacci numbers).

t = Union[Flatten[Table[Fibonacci[n] - Fibonacci[i], {n, 100}, {i, n}]]]; t2 = Select[t, IntegerQ[Sqrt[#]] &]; Sqrt[t2] (* T. D. Noe, Feb 12 2013 *)

A050939 Numbers that are not the sum of consecutive Fibonacci numbers.

Original entry on oeis.org

9, 14, 15, 17, 22, 23, 24, 25, 27, 28, 30, 35, 36, 37, 38, 39, 40, 41, 43, 44, 45, 46, 48, 49, 51, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 77, 78, 79, 80, 82, 83, 85, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101

Offset: 1

N. J. A. Sloane, Jan 02 2000

nonn

From Clark Kimberling, Dec 16 2009: (Start)
(1) This is the ordered sequence of positive numbers that are not the difference between two Fibonacci numbers; see A007298 for a proof.
(2) Let s=(1,2,1,4,2,1,7,4,2,1,12,7,4,2,1,...) be the lengths of runs of consecutive numbers missing from A050939. Is s=A104582? (End)

Cf. A007298, A204922, A204924.

(See A204924, which generates an ordered list of differences of Fibonacci numbers, as in A204922.)

A113191 Difference of two Lucas numbers.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15, 16, 17, 18, 22, 25, 26, 27, 28, 29, 36, 40, 43, 44, 45, 46, 47, 58, 65, 69, 72, 73, 74, 75, 76, 94, 105, 112, 116, 119, 120, 121, 122, 123, 152, 170, 181, 188, 192, 195, 196, 197, 198, 199, 246, 275, 293, 304

Offset: 1

T. D. Noe, Oct 17 2005

nonn

Also the sum of consecutive Lucas numbers because the difference L(i) - L(j) equals the sum L(j+1) + ... + L(i+2).

Conjecture: L(m) - L(n) with m > 1 and m > n >= 0 is a perfect power but not a square only for (m,n) = (7,0), (5,2). This has been verified for n < m <= 500. Note that L(7) - L(0) = 29 - 2 = 3^3 and L(5) - L(2) = 11 - 3 = 2^3. - Zhi-Wei Sun, Jan 02 2025

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

Cf. A000032 (Lucas numbers), A007298 (difference of two Fibonacci numbers).

Cf. A221471, A221472 (square root of squares in this sequence).

Lucas[n_] := Fibonacci[n+1]+Fibonacci[n-1]; Union[Flatten[Table[Lucas[n]-Lucas[i], {n, 13}, {i, 0, n-2}]]]

A129713 Triangle read by rows: T(n,k) is the number of Fibonacci binary words of length n and starting with exactly k 1's (0<=k<=n). A Fibonacci binary word is a binary word having no 00 subword.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 5, 3, 2, 1, 1, 1, 8, 5, 3, 2, 1, 1, 1, 13, 8, 5, 3, 2, 1, 1, 1, 21, 13, 8, 5, 3, 2, 1, 1, 1, 34, 21, 13, 8, 5, 3, 2, 1, 1, 1, 55, 34, 21, 13, 8, 5, 3, 2, 1, 1, 1, 89, 55, 34, 21, 13, 8, 5, 3, 2, 1, 1, 1, 144, 89, 55, 34, 21, 13, 8, 5, 3, 2, 1, 1, 1, 233, 144

Offset: 0

Emeric Deutsch, May 12 2007

Row sums are the Fibonacci numbers (A000045). Sum(k*T(n,k), 0<=k<=n) = F(n+3)-2 = A001911(n).

			T(6,2) = 3 because we have 110110, 110111, 110101.
Triangle starts:
1;
1,1;
1,1,1;
2,1,1,1;
3,2,1,1,1;
5,3,2,1,1,1;
8,5,3,2,1,1,1;

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Cf. A000045, A001911.
Cf. A054123.
Cf. A007298. - Altug Alkan, May 03 2016

a129713 n k = a129713_tabl !! n !! k
a129713_row n = a129713_tabl !! n
a129713_tabl = [1] : [1, 1] : f [1] [1, 1] where
   f us vs = ws : f vs ws where
             ws = zipWith (+) (init us ++ [0, 0, 0]) (vs ++ [1])
-- Reinhard Zumkeller, May 26 2015

with(combinat): T:=proc(n,k) if k<=n-2 then fibonacci(n-k) elif k=n-1 or k=n then 1 else 0 fi end: for n from 0 to 15 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

nn=15;a=1/(1-y x);b=1/(1-x);Map[Select[#,#>0&]&,CoefficientList[Series[a (1+x)/(1-x^2b),{x,0,nn}],{x,y}]]//Grid (* Geoffrey Critzer, Dec 04 2013 *)

T(n,k) = F(n-k) if k<=n-2, T(n,n-1) = T(n,n) = 1, where F(j) are the Fibonacci numbers (F(0)=0, F(1)=1). G.f.: G(t,z) = (1-z^2)/[(1-z-z^2)(1-tz)].

a(n) = A007298(n+4) - A007298(n+3). - Altug Alkan, May 03 2016

A290748 Let F denote the two-way infinite sequence of Fibonacci numbers (for all positive or negative integers k, F(k+2)=F(k)+F(k+1) with F(0)=0, F(1)=1). Sequence lists positive numbers that are the difference between two terms of F.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 18, 19, 20, 21, 22, 23, 24, 26, 29, 31, 32, 33, 34, 35, 37, 42, 47, 50, 52, 53, 54, 55, 56, 57, 58, 60, 63, 68, 76, 81, 84, 86, 87, 88, 89, 90, 92, 97, 110, 123, 131, 136, 139, 141, 142, 143, 144, 145, 146

Offset: 1

N. J. A. Sloane, Aug 11 2017

nonn

			9 is here because F(6) - F(-2) = 8 - (-1) = 9.

Robert Israel, Table of n, a(n) for n = 1..10000
Don Reble, Difference of Fibonacci's, Posting to Sequence Fans Mailing List, Aug 10 2017.

Cf. A000045, A007298 (if we only use F(k) for k >= 0).

See A290749 for the complement.

N:= 40: # to get all terms <= F(N) - F(N-1)
P:= sort(convert({seq(combinat:-fibonacci(n),n=-N..N)},list)):
sort(convert(select(`<=`,{seq(seq(P[i]-P[j],j=1..i-1),i=1..nops(P))},P[-1]-P[-2]),list)): # Robert Israel, Aug 11 2017

Select[Union[Subtract @@@ Tuples[Fibonacci[Range[-30, 30]], 2]], 0 < # < 150 &] (* Giovanni Resta, Aug 11 2017 *)

Corrected by R. J. Mathar, Aug 10 2017

More terms from Giovanni Resta, Aug 11 2017

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

A272631 Sum of three or more consecutive Fibonacci numbers.

Original entry on oeis.org

2, 4, 6, 7, 10, 11, 12, 16, 18, 19, 20, 26, 29, 31, 32, 33, 42, 47, 50, 52, 53, 54, 68, 76, 81, 84, 86, 87, 88, 110, 123, 131, 136, 139, 141, 142, 143, 178, 199, 212, 220, 225, 228, 230, 231, 232, 288, 322, 343, 356, 364, 369, 372, 374, 375, 376, 466, 521, 555, 576

Offset: 1

Altug Alkan, May 04 2016

nonn

Except the first term that is 2, this sequence lists non-Fibonacci numbers (A001690) that are the difference of two Fibonacci numbers. So 2 is the only Fibonacci number in this sequence.

Since the sum of two consecutive Fibonacci numbers is obviously a Fibonacci number because of the definition of Fibonacci numbers, this sequence focuses on the sum of three or more consecutive Fibonacci numbers.

			4 is a term because Fibonacci(1) + Fibonacci(2) + Fibonacci(3) = 1 + 1 + 2 = 4.

Cf. A000045, A001690, A007298.

mx=10^4; i=1; Union@ Reap[ While[(s = Plus @@ Fibonacci[i + {0,1,2}]) <= mx, j = ++i + 1; While[s <= mx, Sow@s; s += Fibonacci@ ++j]]][[2, 1]] (* Giovanni Resta, May 04 2016 *)

A272712 Perfect powers that are the difference of two nonnegative Fibonacci numbers.

Original entry on oeis.org

1, 4, 8, 16, 32, 81, 144, 225, 343, 576

Offset: 1

Altug Alkan, May 05 2016

Listed 10 terms are 1, 2^2, 2^3, 2^4, 2^5, 3^4, 12^2, 15^2, 3^5, 24^2.
1, 4, 8, 16, 32, 81, 343 are also members of A000961.
1, 4, 8, 16, 144 are in the intersection of this sequence and A272575.
Is this sequence finite?
If a(11) exists, it must be larger than 10^2000. - Giovanni Resta, May 25 2016

			32 is a term because 32 = 2^5 = 34 - 2 = Fibonacci(9) - Fibonacci(3).

Cf. A000961, A007298, A001597, A219114, A272575.

isA272712 := proc(n)
    isA001597(n) and isA007298(n) ; #uses code in A001597 and A007298
end proc:
for n from 1 do
    if isA272712(n) then
        printf("%d\n",n) ;
    end if;
end do: # R. J. Mathar, May 25 2016

isA001597[n_] := n == 1 || GCD @@ FactorInteger[n][[All, 2]] > 1;
isA007298[n_] := Module[{i, Fi, j, Fj}, For[i = 0, True, i++, Fi = Fibonacci[i]; For[j = i, True, j++, Fj = Fibonacci[j]; Which[Fj - Fi == n, Return@True, Fj - Fi > n, Break[]]]; Fj := Fibonacci[i + 1]; If[Fj - Fi > n, Return@False]]];
Select[Range[1000], isA001597[#] && isA007298[#]&] (* Jean-François Alcover, Nov 16 2023, after R. J. Mathar in A007298 *)

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A113188 Primes that are the difference of two Fibonacci numbers; primes in A007298.

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A219114 Integers n such that n^2 is the difference of two Fibonacci numbers.

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A050939 Numbers that are not the sum of consecutive Fibonacci numbers.

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A113191 Difference of two Lucas numbers.

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A129713 Triangle read by rows: T(n,k) is the number of Fibonacci binary words of length n and starting with exactly k 1's (0<=k<=n). A Fibonacci binary word is a binary word having no 00 subword.

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A290748 Let F denote the two-way infinite sequence of Fibonacci numbers (for all positive or negative integers k, F(k+2)=F(k)+F(k+1) with F(0)=0, F(1)=1). Sequence lists positive numbers that are the difference between two terms of F.

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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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