A078642 -id:A078642 - cp's OEIS frontend

A118658 a(n) = 2*F(n-1) = L(n) - F(n), where F(n) and L(n) are Fibonacci and Lucas numbers respectively.

2, 0, 2, 2, 4, 6, 10, 16, 26, 42, 68, 110, 178, 288, 466, 754, 1220, 1974, 3194, 5168, 8362, 13530, 21892, 35422, 57314, 92736, 150050, 242786, 392836, 635622, 1028458, 1664080, 2692538, 4356618, 7049156, 11405774, 18454930, 29860704, 48315634, 78176338

Offset: 0

Bill Jones (b92057(AT)yahoo.com), May 18 2006

Essentially the same as A006355, A047992, A054886, A055389, A068922, A078642, A090991. - Philippe Deléham, Sep 20 2006 and Georg Fischer, Oct 07 2018
Also the number of matchings in the (n-2)-pan graph. - Eric W. Weisstein, Jun 30 2016
Also the number of maximal independent vertex sets (and minimal vertex covers) in the (n-1)-ladder graph. - Eric W. Weisstein, Jun 30 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Eric Weisstein's World of Mathematics, Independent Edge Set
Eric Weisstein's World of Mathematics, Ladder Graph
Eric Weisstein's World of Mathematics, Matching
Eric Weisstein's World of Mathematics, Maximal Independent Vertex Set
Eric Weisstein's World of Mathematics, Minimal Vertex Cover
Eric Weisstein's World of Mathematics, Pan Graph
Index entries for linear recurrences with constant coefficients, signature (1,1).

Cf. A000032, A000045, A003714, A212804.

List([0..40],n->2*Fibonacci(n-1)); # Muniru A Asiru, Oct 07 2018

[Lucas(n) - Fibonacci(n): n in [0..40]]; // Vincenzo Librandi, Sep 14 2014

with(combinat): seq(2*fibonacci(n-1),n=0..40); # Muniru A Asiru, Oct 07 2018
a := n -> -2*I^n*ChebyshevU(n-2, -I/2):
seq(simplify(a(n)), n = 0..39);  # Peter Luschny, Dec 03 2023

LinearRecurrence[{1, 1}, {2, 0}, 100] (* Vladimir Joseph Stephan Orlovsky, Jun 05 2011 *)
Table[LucasL[n] - Fibonacci[n], {n, 0, 40}] (* Vincenzo Librandi, Sep 14 2014 *)
Table[2 Fibonacci[n - 1], {n, 0, 20}] (* Eric W. Weisstein, Jun 30 2017 *)
2 Fibonacci[Range[0, 20] - 1] (* Eric W. Weisstein, Jun 30 2017 *)
Subtract @@@ (Through[{LucasL, Fibonacci}[#]] & /@ Range[0, 20]) (* Eric W. Weisstein, Jun 30 2017 *)
CoefficientList[Series[(2 (-1 + x))/(-1 + x + x^2), {x, 0, 20}], x] (* Eric W. Weisstein, Jun 30 2017 *)

a(n)=fibonacci(n-1)<<1 \\ Charles R Greathouse IV, Jun 05 2011

From Philippe Deléham, Sep 20 2006: (Start)
a(0)=2, a(1)=0; for n > 1, a(n) = a(n-1) + a(n-2).
G.f. (2 - 2*x)/(1 - x - x^2).
a(0)=2 and a(n) = 2*A000045(n-1) for n > 0. (End)
a(n) = A006355(n) + 0^n. - M. F. Hasler, Nov 05 2014
a(n) = Lucas(n-2) + Fibonacci(n-2). - Bruno Berselli, May 27 2015
a(n) = 3*Fibonacci(n-2) + Fibonacci(n-5). - Bruno Berselli, Feb 20 2017
a(n) = 2*A212804(n). - Bruno Berselli, Feb 21 2017
E.g.f.: 2*exp(x/2)*(5*cosh(sqrt(5)*x/2) - sqrt(5)*sinh(sqrt(5)*x/2))/5. - Stefano Spezia, Apr 18 2022

More terms from Philippe Deléham, Sep 20 2006

Corrected by T. D. Noe, Nov 01 2006

A090991 Number of meaningful differential operations of the n-th order on the space R^6.

Original entry on oeis.org

6, 10, 16, 26, 42, 68, 110, 178, 288, 466, 754, 1220, 1974, 3194, 5168, 8362, 13530, 21892, 35422, 57314, 92736, 150050, 242786, 392836, 635622, 1028458, 1664080, 2692538, 4356618, 7049156, 11405774, 18454930, 29860704, 48315634, 78176338, 126491972

Offset: 1

Branko Malesevic, Feb 29 2004

Apparently a(n) = A054886(n+2) for n=1..1000. - Georg Fischer, Oct 06 2018

Indranil Ghosh, Table of n, a(n) for n = 1..4772
Tanya Khovanova, Recursive Sequences
Branko J. Malesevic, Some combinatorial aspects of differential operation composition on the space R^n, Univ. Beograd, Publ. Elektrotehn. Fak., Ser. Mat. 9 (1998), 29-33.
Branko J. Malesevic, Some combinatorial aspects of differential operation compositions on space R^n, arXiv:0704.0750 [math.DG], 2007.
Dominika Závacká, Cristina Dalfó, and Miquel Angel Fiol, Integer sequences from k-iterated line digraphs, CEUR: Proc. 24th Conf. Info. Tech. - Appl. and Theory (ITAT 2024) Vol 3792, 156-161. See p. 161, Table 2.
Index entries for linear recurrences with constant coefficients, signature (1,1).

Cf. A054886, A055389, A068922, A090989-A090995.

Essentially the same as A006355, A047992 and A078642.

a:=[6,10];; for n in [3..40] do a[n]:=a[n-1]+a[n-2]; od; a; # Muniru A Asiru, Oct 06 2018

m:=40; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(  2*x*(3+2*x)/(1-x-x^2) )); // G. C. Greubel, Feb 02 2019

NUM := proc(k :: integer) local i,j,n,Fun,Identity,v,A; n := 6; # <- DIMENSION Fun := (i,j)->piecewise(((j=i+1) or (i+j=n+1)),1,0); Identity := (i,j)->piecewise(i=j,1,0); v := matrix(1,n,1); A := piecewise(k>1,(matrix(n,n,Fun))^(k-1),k=1,matrix(n,n,Identity)); return(evalm(v&*A&*transpose(v))[1,1]); end:

CoefficientList[Series[2*(3+2z)/(1-z-z^2), {z, 0, 40}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 11 2011 *)

my(x='x+O('x^40)); Vec(2*x*(3+2*x)/(1-x-x^2)) \\ G. C. Greubel, Feb 02 2019

(2*(3+2*x)/(1-x-x^2)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Feb 02 2019

a(k+4) = 3*a(k+2) - a(k).
a(k) = 2*Fibonacci(k+3).
From Philippe Deléham, Nov 19 2008: (Start)
a(n) = a(n-1) + a(n-2), n>2, where a(1)=6, a(2)=10.
G.f.: 2*x*(3+2*x)/(1-x-x^2). (End)
E.g.f.: 4*exp(x/2)*(5*cosh(sqrt(5)*x/2) + 2*sqrt(5)*sinh(sqrt(5)*x/2))/5 - 4. - Stefano Spezia, Apr 18 2022

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

A068922 Number of ways to tile a 3 X 2n room with 1 X 2 Tatami mats. At most 3 Tatami mats may meet at a point.

Original entry on oeis.org

3, 4, 6, 10, 16, 26, 42, 68, 110, 178, 288, 466, 754, 1220, 1974, 3194, 5168, 8362, 13530, 21892, 35422, 57314, 92736, 150050, 242786, 392836, 635622, 1028458, 1664080, 2692538, 4356618, 7049156, 11405774, 18454930, 29860704, 48315634, 78176338, 126491972

Offset: 1

Dean Hickerson, Mar 11 2002

Colin Barker, Table of n, a(n) for n = 1..1000
R. J. Mathar, Paving Rectangular Regions with Rectangular Tiles: Tatami and Non-Tatami Tilings, arXiv:1311.6135 [math.CO], 2013, Table 2.
F. Ruskey and J. Woodcock, Counting Fixed-Height Tatami Tilings, Electronic Journal of Combinatorics, Paper R126 (2009) 20 pages.
Index entries for linear recurrences with constant coefficients, signature (1,1).

Cf. A068928 for incongruent tilings, A068920 for more info. First column of A272472.
Essentially the same as A006355.
Essentially the same as A078642. - Georg Fischer, Oct 06 2018

Concatenation([3],List([2..40],n->2*Fibonacci(n+1))); # Muniru A Asiru, Oct 07 2018

[3] cat [2*Fibonacci(n+1): n in [2..50]]; // Vincenzo Librandi, Oct 07 2018

with(combinat): 3,seq(2*fibonacci(n+1),n=2..40); # Muniru A Asiru, Oct 07 2018

Join[{3}, Table[2 Fibonacci[n + 1], {n, 2, 50}]] (* Vincenzo Librandi, Oct 07 2018 *)
CoefficientList[Series[(x^2-x-3) / (x^2+x-1), {x, 0, 50}], x] (* Stefano Spezia, Oct 07 2018 *)

Vec(x*(3+x-x^2) / (1-x-x^2) + O(x^50)) \\ Colin Barker, Jan 29 2017

For n >= 2, a(n) = 2*F(n+1), where F(n)=A000045(n) is the n-th Fibonacci number.
G.f.: x*(x^2-x-3) / (x^2+x-1). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009; checked and corrected by R. J. Mathar, Sep 16 2009
From Colin Barker, Jan 29 2017: (Start)
a(n) = (2^(-n)*(-(1-sqrt(5))^(1+n) + (1+sqrt(5))^(1+n))) / sqrt(5) for n>1.
a(n) = a(n-1) + a(n-2) for n>3. (End)
E.g.f.: 2*exp(x/2)*(5*cosh(sqrt(5)*x/2) + sqrt(5)*sinh(sqrt(5)*x/2))/5 - 2 + x. - Stefano Spezia, Apr 18 2022

A154691 Expansion of (1+x+x^2) / ((1-x)*(1-x-x^2)).

Original entry on oeis.org

1, 3, 7, 13, 23, 39, 65, 107, 175, 285, 463, 751, 1217, 1971, 3191, 5165, 8359, 13527, 21889, 35419, 57311, 92733, 150047, 242783, 392833, 635619, 1028455, 1664077, 2692535, 4356615, 7049153, 11405771, 18454927, 29860701, 48315631, 78176335

Offset: 0

R. J. Mathar, Jan 14 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Tomislav Došlić and Biserka Kolarec, On Log-Definite Tempered Combinatorial Sequences, Mathematics (2025) Vol. 13, Iss. 7, 1179.
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Cf. A000045, A001595, A006355, A055389, A066629, A078642, A166863.

a154691 n = a154691_list !! n
a154691_list = 1 : zipWith (+)
                   a154691_list (drop 2 $ map (* 2) a000045_list)
-- Reinhard Zumkeller, Nov 17 2013

A154691:= func< n | 2*Fibonacci(n+3) - 3 >;
[A154691(n): n in [0..40]]; // G. C. Greubel, Jan 18 2025

A154691 := proc(n) coeftayl( (1+x+x^2)/(1-x-x^2)/(1-x),x=0,n) ; end proc:

Fibonacci[Range[3,60]]*2 -3 (* Vladimir Joseph Stephan Orlovsky, Mar 19 2010 *)
CoefficientList[Series[(1 + x + x^2)/((1 - x - x^2)(1 - x)), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 18 2012 *)

Vec((1+x+x^2) / ((1-x-x^2)*(1-x)) + O(x^60)) \\ Colin Barker, Feb 01 2017

def A154691(n): return 2*fibonacci(n+3) - 3
print([A154691(n) for n in range(41)]) # G. C. Greubel, Jan 18 2025

a(n+1) - a(n) = A006355(n+3) = A055389(n+3).
a(n) = A066629(n-1) + A066629(n).
a(n) = A006355(n+4) - 3 = A078642(n+1) - 3.
a(n+1) = a(n) + 2*A000045(n+2). - Reinhard Zumkeller, Nov 17 2013
From Colin Barker, Feb 01 2017: (Start)
a(n) = -3 + (2^(1-n)*((1-r)^n*(-2+r) + (1+r)^n*(2+r))) / r where r=sqrt(5).
a(n) = 2*a(n-1) - a(n-3) for n>2. (End)
a(n) = 2*Fibonacci(n+3) - 3. - Greg Dresden, Oct 10 2020
E.g.f.: 4*exp(x/2)*(5*cosh(sqrt(5)*x/2) + 2*sqrt(5)*sinh(sqrt(5)*x/2))/5 - 3*exp(x). - Stefano Spezia, Apr 09 2025

A163714 Number of n X 2 binary arrays with all 1s connected, a path of 1s from top row to bottom row, and no 1 having more than two 1s adjacent.

Original entry on oeis.org

3, 7, 10, 16, 26, 42, 68, 110, 178, 288, 466, 754, 1220, 1974, 3194, 5168, 8362, 13530, 21892, 35422, 57314, 92736, 150050, 242786, 392836, 635622, 1028458, 1664080, 2692538, 4356618, 7049156, 11405774, 18454930, 29860704, 48315634, 78176338

Offset: 1

R. H. Hardin, Aug 03 2009

nonn

Same recurrence for A163695.

Same recurrence for A163733.

			All solutions for n=4:
...1.0...1.0...1.1...1.1...0.1...0.1...1.1...1.1...1.0...1.1...1.0...1.0...0.1
...1.0...1.0...1.0...1.0...0.1...0.1...0.1...0.1...1.0...1.0...1.1...1.1...0.1
...1.0...1.0...1.0...1.0...0.1...0.1...0.1...0.1...1.1...1.1...0.1...0.1...1.1
...1.0...1.1...1.0...1.1...0.1...1.1...0.1...1.1...0.1...0.1...0.1...1.1...1.0
------
...1.1...0.1...0.1
...0.1...1.1...1.1
...1.1...1.0...1.0
...1.0...1.0...1.1

R. H. Hardin, Table of n, a(n) for n=1..100

Cf. A090991, A078642, A047992. - R. J. Mathar, Aug 06 2009

Empirical: a(n) = a(n-1) + a(n-2) for n>=5.
Conjectures from Colin Barker, Feb 22 2018: (Start)
G.f.: x*(1 + x)*(3 + x - x^2) / (1 - x - x^2).
a(n) = (2^(-n)*((1-sqrt(5))^n*(-3+sqrt(5)) + (1+sqrt(5))^n*(3+sqrt(5)))) / sqrt(5) for n>2.
(End)

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

A343010 Integers k for which there exist three consecutive Fibonacci numbers a, b, and c such that abc = k*(a+b+c).

Original entry on oeis.org

0, 1, 3, 20, 52, 357, 935, 6408, 16776, 114985, 301035, 2063324, 5401852, 37024845, 96932303, 664383888, 1739379600, 11921885137, 31211900499, 213929548580, 560074829380, 3838809989301, 10050135028343, 68884650258840, 180342355680792, 1236084894669817

Offset: 1

Amrit Awasthi, Apr 02 2021

F(n-1)*F(n)*F(n+1) = k(n)*(F(n-1)+F(n)+F(n+1)). This implies that k(n)=(F(n-1)*F(n))/2. Now k(n) will be an integer only when n is of the form 3*m or 3*m+1. Therefore we get k = (F(3*m+-1)*F(3*m))/2.

			0 is a term because F(0)*F(1)*F(2)/(F(0)+F(1)+F(2)) is 0*1*1/(0+1+1) = 0.
1 is a term because F(2)*F(3)*F(4)/(F(2)+F(3)+F(4)) is 1*2*3/(1+2+3) = 1.
3 is a term because F(3)*F(4)*F(5)/(F(3)+F(4)+F(5)) is 2*3*5/(2+3+5) = 3.

Index entries for linear recurrences with constant coefficients, signature (0,17,0,17,0,-1).

Cf. A000045 (Fibonacci numbers), 1/2 times the even terms of sequence A001654.

Cf. A065563 (F(n-1)*F(n)*F(n+1)), A078642 (F(n-1)+F(n)+F(n+1)).

F:= n-> (<<0|1>, <1|1>>^n)[1, 2]:
a:= n-> (k-> mul(F(k+j), j=0..2)/add(F(k+j), j=0..2))(floor(3*n/2)-1):
seq(a(n), n=1..30);  # Alois P. Heinz, Apr 02 2021

Select[Table[(Fibonacci[k-1]*Fibonacci[k]*Fibonacci[k+1])/(Fibonacci[k-1]+Fibonacci[k]+Fibonacci[k+1]),{k,37}],IntegerQ] (* or *)
b[k_]:=Fibonacci[3k-1]*Fibonacci[3k]/2; c[k_]:=Fibonacci[3k+1]*Fibonacci[3k]/2; Union[Table[b[k],{k,0,12}],Table[c[k],{k,0,12}]] (* Stefano Spezia, Apr 03 2021 *)

r(m)={fibonacci(m)*fibonacci(m-1)*fibonacci(m+1)/(fibonacci(m)+fibonacci(m-1)+fibonacci(m+1))}
{ for(m=2, 30, my(t=r(m)); if(!frac(t), print1(t, ", ")))} \\ Andrew Howroyd, Apr 02 2021

Union of the two sequences b(k) and c(k) defined respectively as F(3*k-1)*F(3*k)/2 and F(3*k+1)*F(3*k)/2.

G.f.: x^2*(1 + 3*x + 3*x^2 + x^3)/(1 - 17*x^2 - 17*x^4 + x^6). - Stefano Spezia, Apr 03 2021

cp's OEIS Frontend

A118658 a(n) = 2*F(n-1) = L(n) - F(n), where F(n) and L(n) are Fibonacci and Lucas numbers respectively.

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A090991 Number of meaningful differential operations of the n-th order on the space R^6.

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A059389 Sums of two nonzero Fibonacci numbers.

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A068922 Number of ways to tile a 3 X 2n room with 1 X 2 Tatami mats. At most 3 Tatami mats may meet at a point.

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Magma

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A154691 Expansion of (1+x+x^2) / ((1-x)*(1-x-x^2)).

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A163714 Number of n X 2 binary arrays with all 1s connected, a path of 1s from top row to bottom row, and no 1 having more than two 1s adjacent.

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A343010 Integers k for which there exist three consecutive Fibonacci numbers a, b, and c such that abc = k*(a+b+c).