A119457 -id:A119457 - cp's OEIS frontend

A006355 Number of binary vectors of length n containing no singletons.

1, 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

Offset: 0

David M. Bloom

Number of cvtemplates at n-2 letters given <= 2 consecutive consonants or vowels (n >= 4).
Number of (n,2) Freiman-Wyner sequences.
Diagonal sums of the Riordan array ((1-x+x^2)/(1-x), x/(1-x)), A072405 (where this begins 1,0,1,1,1,1,...). - Paul Barry, May 04 2005
Central terms of the triangle in A094570. - Reinhard Zumkeller, Mar 22 2011
Pisano period lengths: 1, 1, 8, 3, 20, 8, 16, 6, 24, 20, 10, 24, 28, 16, 40, 12, 36, 24, 18, 60, ... . - R. J. Mathar, Aug 10 2012
Also the number of matchings in the (n-2)-pan graph for n >= 5. - Eric W. Weisstein, Oct 03 2017
a(n) is the number of bimultus bitstrings of length n. A bitstring is bimultus if each of its 1's possess at least one neighboring 1 and each of its 0's possess at least one neighboring 0. - Steven Finch, May 26 2020

			a(6)=10 because we have: 000000, 000011, 000111, 001100, 001111, 110000, 110011, 111000, 111100, 111111. - _Geoffrey Critzer_, Jan 26 2014

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 16, 51.

Charles R Greathouse IV, Table of n, a(n) for n = 0..4786 (next term has 1001 digits)
Kassie Archer and Aaron Geary, Powers of permutations that avoid chains of patterns, arXiv:2312.14351 [math.CO], 2023. See p. 15.
J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. [Annotated scanned copy]
Ian F. Blake, The enumeration of certain run length sequences, Information and Control, 55 (1982), 222-237.
Alexander Burstein, Sergey Kitaev, and Toufik Mansour, Partially ordered patterns and their combinatorial interpretations, PU. M. A. Vol. 19 (2008), No. 2-3, pp. 27-38.
Steven Finch, Variance of longest run duration in a random bitstring, arXiv:2005.12185 [math.CO], 2020.
Enoch Haga, Room for Expansion, Word Ways, 33 (No. 2, 2000), pp. 106-113 (see p. 110).
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 898.
Sergey Kitaev and Jeffrey Remmel, (a,b)-rectangle patterns in permutations and words, arXiv:1304.4286 [math.CO], 2013.
Thor Martinsen, Non-Fisherian generalized Fibonacci numbers, Notes Num. Theor. Disc. Math. (2025) Vol. 31, No. 2, 370-389. See p. 389.
Noriaki Sannomiya, Hosho Katsura, and Yu Nakayama, Supersymmetry breaking and Nambu-Goldstone fermions with cubic dispersion, arXiv preprint arXiv:1612.02285 [cond-mat.str-el], 2016-2017. See Table II, line 2.
Eric Weisstein's World of Mathematics, Independent Edge Set, Matching and Pan Graph.
Index entries for linear recurrences with constant coefficients, signature (1,1).

Except for initial term, = 2*Fibonacci numbers (A000045).
Essentially the same as A047992, A054886, A055389, A068922, and A090991.
Column 2 in A265584.
Cf. A094570, A097925, A097926, A118658, A119457.

a006355 n = a006355_list !! n
a006355_list = 1 : fib2s where
   fib2s = 0 : map (+ 1) (scanl (+) 1 fib2s)
-- Reinhard Zumkeller, Mar 20 2013

[1] cat [Lucas(n) - Fibonacci(n): n in [1..50]]; // Vincenzo Librandi, Aug 02 2014

a:= n-> if n=0 then 1 else (Matrix([[2,-2]]). Matrix([[1,1], [1,0]])^n) [1,1] fi: seq(a(n), n=0..38); # Alois P. Heinz, Aug 18 2008
a := n -> ifelse(n=0, 1, -2*I^n*ChebyshevU(n-2, -I/2)):
seq(simplify(a(n)), n = 0..38);  # Peter Luschny, Dec 03 2023

Join[{1}, Last[#] - First[#] & /@ Partition[Fibonacci[Range[-1, 40]], 4, 1]] (* Harvey P. Dale, Sep 30 2011 *)
Join[{1}, LinearRecurrence[{1, 1}, {0, 2}, 38]] (* Jean-François Alcover, Sep 23 2017 *)
(* Programs from Eric W. Weisstein, Oct 03 2017 *)
Join[{1}, Table[2 Fibonacci[n], {n, 0, 40}]]
Join[{1}, 2 Fibonacci[Range[0, 40]]]
CoefficientList[Series[(1-x+x^2)/(1-x-x^2), {x, 0, 40}], x] (* End *)

a(n)=if(n,2*fibonacci(n-1),1) \\ Charles R Greathouse IV, Mar 14 2012

my(x='x+O('x^50)); Vec((1-x+x^2)/(1-x-x^2)) \\ Altug Alkan, Nov 01 2015

def A006355(n): return 2*fibonacci(n-1) - int(n==0)
print([A006355(n) for n in range(51)]) # G. C. Greubel, Apr 18 2025

a(n+2) = F(n-1) + F(n+2), for n > 0.
G.f.: (1-x+x^2)/(1-x-x^2). - Paul Barry, May 04 2005
a(n) = A119457(n-1,n-2) for n > 2. - Reinhard Zumkeller, May 20 2006
a(n) = 2*F(n-1) for n > 0, F(n)=A000045(n) and a(0)=1. - Mircea Merca, Jun 28 2012
G.f.: 1 - x + x*Q(0), where Q(k) = 1 + x^2 + (2*k+3)*x - x*(2*k+1 + x)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 05 2013
a(n) = A118658(n) - 0^n. - M. F. Hasler, Nov 05 2014
a(n) = 2^(-n)*((1+r)*(1-r)^n - (1-r)*(1+r)^n)/r for n > 0, where r=sqrt(5). - Colin Barker, Jan 28 2017
a(n) = a(n-1) + a(n-2) for n >= 3. - Armend Shabani, Nov 25 2020
E.g.f.: 2*exp(x/2)*(5*cosh(sqrt(5)*x/2) - sqrt(5)*sinh(sqrt(5)*x/2))/5 - 1. - Stefano Spezia, Apr 18 2022
a(n) = F(n-3) + F(n-2) + F(n-1) for n >= 3, where F(n)=A000045(n). - Gergely Földvári, Aug 03 2024

Corrected by T. D. Noe, Oct 31 2006

A001891 Hit polynomials; convolution of natural numbers with Fibonacci numbers F(2), F(3), F(4), ....

Original entry on oeis.org

0, 1, 4, 10, 21, 40, 72, 125, 212, 354, 585, 960, 1568, 2553, 4148, 6730, 10909, 17672, 28616, 46325, 74980, 121346, 196369, 317760, 514176, 831985, 1346212, 2178250, 3524517, 5702824, 9227400, 14930285, 24157748, 39088098, 63245913, 102334080, 165580064

Offset: 0

N. J. A. Sloane, Simon Plouffe

a(n) is the sum of the n-th row of the triangle in A119457 for n > 0. - Reinhard Zumkeller, May 20 2006
Convolution of odds (A005408) with Fibonacci numbers (A000045). - Graeme McRae, Jun 06 2006
Equals row sums of triangle A152203. - Gary W. Adamson, Nov 29 2008
Define a triangle by T(n,0) = n*(n+1)+1, T(n,n) = 1, and T(r,c) = T(r-1,c) + T(r-2,c-1). This triangle starts: 1; 3,1; 7,2,1; 13,5,2,1; 21,12,4,2,1; the sum of terms in row n is a(n+1). - J. M. Bergot, Apr 23 2013
a(n) = number of k-tuples (u(1), u(2), ..., u(k)) with 1 <= u(1) < u(2) < ... < u(k) <= n such that u(i) - u(i-1) <= 2 for i = 2,...,k. Changing the bound from 2 to 3, then 4, then 5, yields A356619, A356620, A356621. The patterns suggest that the limiting sequence as the bound increases is A000295. - Clark Kimberling, Aug 24 2022

J. Riordan, The enumeration of permutations with three-ply staircase restrictions, unpublished memorandum, Bell Telephone Laboratories, Murray Hill, NJ, Oct 1963. (See A001883)
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
N. J. A. Sloane, Annotated copy of Riordan's Three-Ply Staircase paper (unpublished memorandum, Bell Telephone Laboratories, Murray Hill, NJ, Oct 1963)
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Partial sums of A001911.
A diagonal of triangle in A080061.
Right-hand column 5 of triangle A011794.
Cf. A001883-A001890. A152203.
Cf. A000295, A356619, A356620, A356621.

List([0..40], n-> Fibonacci(n+5) -2*n-5); # G. C. Greubel, Jul 06 2019

[Fibonacci(n+5)-(5+2*n): n in [0..40]]; // Vincenzo Librandi, Jun 07 2013

LinearRecurrence[{3,-2,-1,1}, {0,1,4,10}, 40] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2012 *)
Table[Fibonacci[n+5] -(2*n+5), {n,0,40}] (* G. C. Greubel, Jul 06 2019 *)
maxDiff = 2;
Map[Length[Select[Map[{#, Max[Differences[#]]} &,
  Drop[Subsets[Range[#]], # + 1]], #[[2]] <= maxDiff &]] &,
  Range[16]] (* Peter J. C. Moses, Aug 14 2022 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; 1,-1,-2,3]^n*[0;1;4;10])[1,1] \\ Charles R Greathouse IV, Apr 08 2016

[fibonacci(n+5) -2*n-5 for n in (0..40)] # G. C. Greubel, Jul 06 2019

G.f.: x*(1+x)/((1-x-x^2)*(1-x)^2). - Simon Plouffe in his 1992 dissertation
a(n) = Fibonacci(n+5) - (5+2*n). - Wolfdieter Lang
a(n) = a(n-1) + a(n-2) + (2n+1); a(-x)=0. - Barry E. Williams, Mar 27 2000
a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4). - Sam Lachterman (slachterman(AT)fuse.net), Sep 22 2003
a(n) - a(n-1) = A101220(2,1,n). - Ross La Haye, May 31 2006
a(n) = (-3 + (2^(-1-n)*((1-sqrt(5))^n*(-11+5*sqrt(5)) + (1+sqrt(5))^n*(11+5*sqrt(5)))) / sqrt(5) - 2*(1+n)). - Colin Barker, Mar 11 2017

A022086 Fibonacci sequence beginning 0, 3.

Original entry on oeis.org

0, 3, 3, 6, 9, 15, 24, 39, 63, 102, 165, 267, 432, 699, 1131, 1830, 2961, 4791, 7752, 12543, 20295, 32838, 53133, 85971, 139104, 225075, 364179, 589254, 953433, 1542687, 2496120, 4038807, 6534927, 10573734, 17108661, 27682395, 44791056, 72473451, 117264507

Offset: 0

N. J. A. Sloane

First differences of A111314. - Ross La Haye, May 31 2006
Pisano period lengths: 1, 3, 1, 6, 20, 3, 16, 12, 8, 60, 10, 6, 28, 48, 20, 24, 36, 24, 18, 60, ... . - R. J. Mathar, Aug 10 2012
For n>=6, a(n) is the number of edge covers of the union of two cycles C_r and C_s, r+s=n, with a single common vertex. - Feryal Alayont, Oct 17 2024

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 7,17.

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

Essentially the same as A097135.
Cf. A026390, A036999, A111314, A119457, A187893.
Sequences of the form Fibonacci(n+k) + Fibonacci(n-k) are listed in A280154.
Sequences of the form m*Fibonacci: A000045 (m=1), A006355 (m=2), this sequence (m=3), A022087 (m=4), A022088 (m=5), A022089 (m=6), A022090 (m=7), A022091 (m=8), A022092 (m=8), A022093 (m=10), A022345...A022366 (m=11...32).

[3*Fibonacci(n): n in [0..40]]; // Vincenzo Librandi, Dec 31 2016

BB := n->if n=0 then 3; > elif n=1 then 0; > else BB(n-2)+BB(n-1); > fi: > L:=[]: for k from 1 to 34 do L:=[op(L),BB(k)]: od: L; # Zerinvary Lajos, Mar 19 2007
with (combinat):seq(sum((fibonacci(n,1)),m=1..3),n=0..32); # Zerinvary Lajos, Jun 19 2008

LinearRecurrence[{1, 1}, {0, 3}, 40] (* Arkadiusz Wesolowski, Aug 17 2012 *)
Table[Fibonacci[n + 4] + Fibonacci[n - 4] - 4 Fibonacci[n], {n, 0, 40}] (* Bruno Berselli, Dec 30 2016 *)
Table[3 Fibonacci[n], {n, 0, 40}] (* Vincenzo Librandi, Dec 31 2016 *)

a(n)=3*fibonacci(n) \\ Charles R Greathouse IV, Nov 06 2014

def A022086(n): return 3*fibonacci(n)
print([A022086(n) for n in range(41)]) # G. C. Greubel, Apr 10 2025

a(n) = 3*Fibonacci(n).
a(n) = F(n-2) + F(n+2) for n>1, with F=A000045.
a(n) = round( ((6*phi-3)/5) * phi^n ) for n>2. - Thomas Baruchel, Sep 08 2004
a(n) = A119457(n+1,n-1) for n>1. - Reinhard Zumkeller, May 20 2006
G.f.: 3*x/(1-x-x^2). - Philippe Deléham, Nov 19 2008
a(n) = A187893(n) - 1. - Filip Zaludek, Oct 29 2016
E.g.f.: 6*sinh(sqrt(5)*x/2)*exp(x/2)/sqrt(5). - Ilya Gutkovskiy, Oct 29 2016
a(n) = F(n+4) + F(n-4) - 4*F(n), F = A000045. - Bruno Berselli, Dec 29 2016

A023607 a(n) = n * Fibonacci(n+1).

Original entry on oeis.org

0, 1, 4, 9, 20, 40, 78, 147, 272, 495, 890, 1584, 2796, 4901, 8540, 14805, 25552, 43928, 75258, 128535, 218920, 371931, 630454, 1066464, 1800600, 3034825, 5106868, 8580897, 14398412, 24129160, 40388070, 67527579, 112786496, 188195271

Offset: 0

Clark Kimberling

Convolution of Fibonacci numbers and Lucas numbers.
Central terms of the triangle in A119457 for n>0. - Reinhard Zumkeller, May 20 2006
d/dx(1 + x + 2x^2 + 3x^3 + 5x^4 + 8x^5 + ...) = (1 + 4x + 9x^2 + ...). - Gary W. Adamson, Jun 27 2009
For n > 0: sums of rows of the triangle in A108035. - Reinhard Zumkeller, Oct 08 2012

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
M. Griffiths, A Restricted Random Walk defined via a Fibonacci Process, Journal of Integer Sequences, Vol. 14 (2011), #11.5.4.
Milan Janjić, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010) # 10.7.8, section 3.
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

First differences of A094584.
Second column of triangle A016095.
Cf. A000045, A104796.

a023607 n = a023607_list !! n
a023607_list = zipWith (*) [0..] $ tail a000045_list
-- Reinhard Zumkeller, Oct 08 2012

A023607 := proc(n)
    n*combinat[fibonacci](n+1) ;
end proc:
seq(A023607(n),n=0..10) ; # R. J. Mathar, Jul 15 2017

Times@@@Thread[{Range[0, 50], Fibonacci[Range[51]]}]  (* Harvey P. Dale, Mar 08 2011 *)
Table[n*Fibonacci[n + 1], {n, 0, 50}]

a(n)=n*fibonacci(n+1) \\ Charles R Greathouse IV, Sep 24 2015

O.g.f.: x(2x+1)/(1-x-x^2)^2. - Len Smiley, Dec 11 2001
a(n) = n*Sum_{k=0..n} binomial(k,n-k). - Paul Barry, Sep 25 2004
a(n) = A215082(2n-2) + A215082(2n-1). - Philippe Deléham, Aug 03 2012
a(n) = Sum_{i=1..n} A000045(i)*A000032(n-i+1). - Vladimir Kruchinin, Nov 08 2013

Simpler description from Samuel Lachterman (slachterman(AT)fuse.net), Sep 19 2003

Name improved by T. D. Noe, Mar 08 2011

A022087 Fibonacci sequence beginning 0, 4.

Original entry on oeis.org

0, 4, 4, 8, 12, 20, 32, 52, 84, 136, 220, 356, 576, 932, 1508, 2440, 3948, 6388, 10336, 16724, 27060, 43784, 70844, 114628, 185472, 300100, 485572, 785672, 1271244, 2056916, 3328160, 5385076, 8713236, 14098312, 22811548, 36909860, 59721408, 96631268

Offset: 0

N. J. A. Sloane

For n > 1, this sequence gives the number of binary strings of length n that do not contain 0000, 0101, 1010, or 1111 as contiguous substrings (see A230127). - Nathaniel Johnston, Oct 11 2013

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 18.

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

Cf. A000045, A119457.
Cf. similar sequences listed in A258160.
Cf. sequences of the form m*Fibonacci listed in A022086.

[4*Fibonacci(n): n in [0..40]]; // Vincenzo Librandi, Oct 12 2013

a:= n-> (Matrix([[4,0]]). Matrix([[1,1],[1,0]])^n)[1,2]: seq(a(n), n=0..40); # Alois P. Heinz, Aug 17 2008

4*Fibonacci[Range[0,50]] (* Vladimir Joseph Stephan Orlovsky, Jul 22 2008 *)
Table[4 Fibonacci(n), {n, 0, 40}] (* Bruno Berselli, May 22 2015 *)
LinearRecurrence[{1,1},{0,4},40] (* Harvey P. Dale, Jul 31 2025 *)

a(n)=4*fibonacci(n) \\ Charles R Greathouse IV, Jun 05 2011

def A022087(n): return 4*fibonacci(n)
print([A022087(n) for n in range(41)]) # G. C. Greubel, Apr 12 2025

a(n) = 4*F(n) = F(n-2) + F(n) + F(n+2), where F = A000045.
a(n) = round( phi^n*(8*phi-4)/5 ) for n>2. - Thomas Baruchel, Sep 08 2004
a(n) = A119457(n+2,n-1) for n>1. - Reinhard Zumkeller, May 20 2006
G.f.: 4*x/(1-x-x^2). - Philippe Deléham, Nov 19 2008
a(n) = F(n+9) - 17*F(n+3), where F=A000045. - Manuel Valdivia, Dec 15 2009
G.f.: Q(0) -1, where Q(k) = 1 + x^2 + (4*k+5)*x - x*(4*k+1 + x)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 07 2013
a(n) = Fibonacci(n+3) - Fibonacci(n-3), where Fibonacci(-3..-1) = 2,-1,1. - Bruno Berselli, May 22 2015

A022088 Fibonacci sequence beginning 0, 5.

Original entry on oeis.org

0, 5, 5, 10, 15, 25, 40, 65, 105, 170, 275, 445, 720, 1165, 1885, 3050, 4935, 7985, 12920, 20905, 33825, 54730, 88555, 143285, 231840, 375125, 606965, 982090, 1589055, 2571145, 4160200, 6731345, 10891545, 17622890, 28514435, 46137325, 74651760, 120789085

Offset: 0

N. J. A. Sloane

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, pp. 15, 34, 52.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Kristina Lund, Steven Schlicker and Patrick Sigmon, Fibonacci sequences and the space of compact sets, Involve, 1:2 (2008), pp. 159-165.
Index entries for linear recurrences with constant coefficients, signature (1,1).

Cf. A000032, A000045, A000108, A014217, A119457, A267633.

Cf. sequences of the form m*Fibonacci listed in A022086.

[5*Fibonacci(n): n in [1..40]]; // Vincenzo Librandi, Sep 03 2015

LinearRecurrence[{1,1},{0,5},40] (* Harvey P. Dale, Jan 13 2012 *)
5*Fibonacci[Range[0, 50]] (* G. C. Greubel, Feb 10 2023 *)

a(n) = 5*fibonacci(n); \\ Michel Marcus, Sep 03 2015

[5*fibonacci(n) for n in range(51)] # G. C. Greubel, Feb 10 2023

a(n) = round( (2*phi-1)*phi^n ) for n>3. - Thomas Baruchel, Sep 08 2004
a(n) = 5*Fibonacci(n).
a(n) = A119457(n+3,n-1) for n>1. - Reinhard Zumkeller, May 20 2006
G.f.: 5*x/(1-x-x^2). - Philippe Deléham, Nov 20 2008
a(n+2) = A014217(n+4) - A014217(n). - Paul Curtz, Dec 22 2008
a(n) = sqrt(5*(A000032(n)^2 - 4*(-1)^n)). - Alexander Samokrutov, Sep 02 2015
From Tom Copeland, Jan 25 2016: (Start)
The o.g.f. for the shifted series b(0)=0 and b(n) = a(n+1) is G(x) = 5*x*(1+x)/(1-x*(1+x)) = 5 L(-Cinv(-x)), where L(x) = x/(1-x) with inverse Linv(x) = x/(1+x) and Cinv(x) = x*(1-x), the inverse of the o.g.f. for the shifted Catalan numbers of A000108, C(x) = (1-sqrt(1-4*x))/2. Then Ginv(x) = -C(-Linv(x/5)) = (-1 + sqrt(1+4*x/(5+x)))/2.
a(n+1) = 5*Sum_{k=0..n} binomial(n-k,k) = 5 * A000045(n+1), from A267633, with the convention for zeros of the binomial assumed there. (End)
For n > 0, 1/a(n) = Sum_{k>=1} F(n*k)/(L(n+1)^(k+1)), where F(n) = A000045(n) and L(n) = A000032(n). - Diego Rattaggi, Oct 26 2022

A022089 Fibonacci sequence beginning 0, 6.

Original entry on oeis.org

0, 6, 6, 12, 18, 30, 48, 78, 126, 204, 330, 534, 864, 1398, 2262, 3660, 5922, 9582, 15504, 25086, 40590, 65676, 106266, 171942, 278208, 450150, 728358, 1178508, 1906866, 3085374, 4992240, 8077614, 13069854, 21147468, 34217322, 55364790, 89582112, 144946902

Offset: 0

N. J. A. Sloane

Starting with a(0)=1, a(1)=3, a(n) = the number of ternary length-2 squarefree words of length n.

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 15.

G. C. Greubel, Table of n, a(n) for n = 0..1000
N. H. Bong, C. Dalfó, M. À. Fiol, and D. Závacká, Some inner metric parameters of a digraph: Iterated line digraphs and integer sequences, arXiv:2409.02125 [math.CO], 2024. See p. 17.
Cristina Dalfó and Miquel Àngel Fiol, A Note on the Order of Iterated Line Digraphs, Journal of Graph Theory, Volume 85, Issue 2, June 2017, Pages 395-39, 2016; DOI: 10.1002/jgt.22068; arXiv:1607.08832 [math.CO], 2016.
Tanya Khovanova, Recursive Sequences
Christoph Richard and Uwe Grimm, On the entropy and letter frequencies of ternary squarefree words, arXiv:math/0302302 [math.CO], 2003.
Index entries for linear recurrences with constant coefficients, signature (1,1).

Cf. A000032, A000045, A119457.

Sequences of the form m*Fibonacci listed in A022086.

A022089:= func< n | 6*Fibonacci(n) >;
[A022089(n): n in [0..50]]; // G. C. Greubel, Apr 13 2025

a:= n-> 6*(<<0|1>, <1|1>>^n)[1,2]:
seq(a(n), n=0..40);  # Alois P. Heinz, Jan 18 2019

6*Fibonacci[Range[0,50]] (* G. C. Greubel, Apr 13 2025 *)
LinearRecurrence[{1,1},{0,6},50] (* Harvey P. Dale, Dec 05 2015 *)

def A022089(n): return 6*fibonacci(n)
print([A022089(n) for n in range(51)]) # G. C. Greubel, Apr 13 2025

a(n) = round( (12*phi-6)/5 * phi^n)  for n>3. - Thomas Baruchel, Sep 08 2004
a(n) = 6*F(n) = F(n+3) + F(n+1) + F(n-4), n>3, where F=A000045.
a(n) = A119457(n+4,n-1) for n>1. - Reinhard Zumkeller, May 20 2006
G.f.: 6*x/(1-x-x^2). - Philippe Deléham, Nov 20 2008
a(n) = 6 * A000045(n). - Alois P. Heinz, Jan 18 2019

A022090 Fibonacci sequence beginning 0, 7.

Original entry on oeis.org

0, 7, 7, 14, 21, 35, 56, 91, 147, 238, 385, 623, 1008, 1631, 2639, 4270, 6909, 11179, 18088, 29267, 47355, 76622, 123977, 200599, 324576, 525175, 849751, 1374926, 2224677, 3599603, 5824280, 9423883, 15248163, 24672046, 39920209, 64592255, 104512464

Offset: 0

N. J. A. Sloane

The number of heptagons in the n-th ring of the Klein Quartic. - Amiram Eldar, Nov 14 2023

Arthur T. Benjamin and Jennifer J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A., 2003, p. 15.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences.
William P. Thurston, The Eightfold Way: A Mathematical Sculpture by Helaman Ferguson, in: The Eightfold Way: The Beauty of the Klein Quartic (ed. Silvio Levy), Cambridge University Press, New York, 1999, pp. 1-7.
Eric Weisstein's World of Mathematics, Klein Quartic.
Index entries for linear recurrences with constant coefficients, signature (1,1).

Cf. A000032, A000045, A001622, A119457.
Sequences of the form Fibonacci(n+k) + Fibonacci(n-k) are listed in A280154.
Sequences of the form m*Fibonacci are listed in A022086.

A022090:= func< n | 7*Fibonacci(n) >; [A022090(n): n in [0..40]]; // G. C. Greubel, Apr 10 2025

7*Fibonacci[Range[0,40]] (* Vladimir Joseph Stephan Orlovsky, Jul 23 2008 *)

def A022090(n): return 7*fibonacci(n)
[A022090(n) for n in range(41)] # G. C. Greubel, Apr 10 2025

a(n) = round(((14*phi-7)/5) * phi^n), for n>3. - Thomas Baruchel, Sep 08 2004
a(n) = 7*Fibonacci(n) = Fibonacci(n+4) + Fibonacci(n-4) for n>3.
a(n) = A119457(n+5, n-1) for n>1. - Reinhard Zumkeller, May 20 2006
G.f.: 7*x/(1-x-x^2). - Philippe Deléham, Nov 20 2008

cp's OEIS Frontend

A006355 Number of binary vectors of length n containing no singletons.

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A001891 Hit polynomials; convolution of natural numbers with Fibonacci numbers F(2), F(3), F(4), ....

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A022086 Fibonacci sequence beginning 0, 3.

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A023607 a(n) = n * Fibonacci(n+1).

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A022087 Fibonacci sequence beginning 0, 4.

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A022088 Fibonacci sequence beginning 0, 5.