A047992 -id:A047992 - 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

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

Original entry on oeis.org

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

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)

cp's OEIS Frontend

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

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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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GAP

Magma

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Mathematica

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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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