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 A002605 a(n) = 2*(a(n-1) + a(n-2)), a(0) = 0, a(1) = 1. 120

%I

%S 0,1,2,6,16,44,120,328,896,2448,6688,18272,49920,136384,372608,

%T 1017984,2781184,7598336,20759040,56714752,154947584,423324672,

%U 1156544512,3159738368,8632565760,23584608256,64434348032,176037912576,480944521216,1313964867584

%N a(n) = 2*(a(n-1) + a(n-2)), a(0) = 0, a(1) = 1.

%C Individually, both this sequence and A028859 are convergents to 1 + sqrt(3). Mutually, both sequences are convergents to 2 + sqrt(3) and 1 + sqrt(3)/2. - Klaus E. Kastberg (kastberg(AT)hotkey.net.au), Nov 04 2001

%C The number of (s(0), s(1), ..., s(n+1)) such that 0 < s(i) < 6 and |s(i) - s(i-1)| <= 1 for i = 1, 2, ..., n + 1, s(0) = 2, s(n+1) = 3. - _Herbert Kociemba_, Jun 02 2004

%C The same sequence may be obtained by the following process. Starting a priori with the fraction 1/1, the denominators of fractions built according to the rule: add top and bottom to get the new bottom, add top and 4 times the bottom to get the new top. The limit of the sequence of fractions is sqrt(4). - _Cino Hilliard_, Sep 25 2005

%C The Hankel transform of this sequence is [1, 2, 0, 0, 0, 0, 0, 0, 0, ...]. - _Philippe Deléham_, Nov 21 2007

%C [1, 3; 1, 1]^n *[1, 0] = [A026150(n), a(n)]. - _Gary W. Adamson_, Mar 21 2008

%C (1 + sqrt(3))^n = A026150(n) + a(n)*sqrt(3). - _Gary W. Adamson_, Mar 21 2008

%C a(n+1) is the number of ways to tile a board of length n using red and blue tiles of length one and two. - _Geoffrey Critzer_, Feb 07 2009

%C Starting with offset 1 = INVERT transform of the Jacobsthal sequence, A001045: (1, 1, 3, 5, 11, 21, ...). - _Gary W. Adamson_, May 12 2009

%C Starting with "1" = INVERTi transform of A007482: (1, 3, 11, 39, 139, ...). - _Gary W. Adamson_, Aug 06 2010

%C An elephant sequence, see A175654. For the corner squares four A vectors, with decimal values 85, 277, 337 and 340, lead to this sequence (without the leading 0). For the central square these vectors lead to the companion sequence A026150, without the first leading 1. - _Johannes W. Meijer_, Aug 15 2010

%C The sequence 0, 1, -2, 6, -16, 44, -120, 328, -896, ... (with alternating signs) is the Lucas U(-2,-2)-sequence. - _R. J. Mathar_, Jan 08 2013

%C a(n+1) counts n-walks (closed) on the graph G(1-vertex;1-loop,1-loop,2-loop,2-loop). - _David Neil McGrath_, Dec 11 2014

%C Number of binary strings of length 2*n - 2 in the regular language (00+11+0101+1010)*. - _Jeffrey Shallit_, Dec 14 2015

%C For n >= 1, a(n) equals the number of words of length n - 1 over {0, 1, 2, 3} in which 0 and 1 avoid runs of odd lengths. - _Milan Janjic_, Dec 17 2015

%C a(n+1) is the number of compositions of n into parts 1 and 2, both of two kinds. - _Gregory L. Simay_, Sep 20 2017

%C Number of associative, quasitrivial, and order-preserving binary operations on the n-element set {1, ..., n} that have neutral elements. - _J. Devillet_, Sep 28 2017

%C (1 + sqrt(3))^n = A026150(n) + a(n)*sqrt(3), for n >= 0; integers in the real quadratic number field Q(sqrt(3)). - _Wolfdieter Lang_, Feb 10 2018

%D John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, p. 16.

%H Vincenzo Librandi, <a href="/A002605/b002605.txt">Table of n, a(n) for n = 0..500</a>

%H A. Abdurrahman, <a href="https://arxiv.org/abs/1909.10889">CM Method and Expansion of Numbers</a>, arXiv:1909.10889 [math.NT], 2019.

%H Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Szczyrba/sz3.html">Analytic Representations of the n-anacci Constants and Generalizations Thereof</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.

%H M. Couceiro, J. Devillet, and J.-L. Marichal, <a href="http://arxiv.org/abs/1709.09162">Quasitrivial semigroups: characterizations and enumerations</a>, arXiv:1709.09162 [math.RA], 2017.

%H M. Diepenbroek, M. Maus, A. Stoll, <a href="http://www.valpo.edu/mathematics-statistics/files/2014/09/Pudwell2015.pdf">Pattern Avoidance in Reverse Double Lists</a>, Preprint 2015. See Table 3.

%H Alice L. L. Gao, Sergey Kitaev, <a href="https://arxiv.org/abs/1903.08946">On partially ordered patterns of length 4 and 5 in permutations</a>, arXiv:1903.08946 [math.CO], 2019.

%H Dale Gerdemann <a href="http://www.youtube.com/watch?v=7fUghbI1y3o">Bird Flock</a>

%H A. F. Horadam, <a href="http://www.fq.math.ca/Scanned/5-5/horadam.pdf">Special properties of the sequence W_n(a,b; p,q)</a>, Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=q=2.

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=476">Encyclopedia of Combinatorial Structures 476</a>

%H D. Jhala, G. P. S. Rathore, K. Sisodiya, <a href="http://dx.doi.org/10.12691/tjant-2-4-3">Some Properties of k-Jacobsthal Numbers with Arithmetic Indexes</a>, Turkish Journal of Analysis and Number Theory, 2014, Vol. 2, No. 4, 119-124.

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H W. Lang, <a href="http://www.fq.math.ca/Scanned/38-5/lang.pdf">On polynomials related to powers of the generating function of Catalan's numbers</a>, Fib. Quart. 38,5 (2000) 408-419; Eqs. (39), (41) and (45), lhs, m=2.

%H D. H. Lehmer, <a href="https://doi.org/10.1112/jlms/s1-10.2.162">On Lucas's test for the primality of Mersenne's numbers</a>, Journal of the London Mathematical Society 1.3 (1935): 162-165. See U_n.

%H Alan Prince, <a href="http://roa.rutgers.edu/article/view/1127">Counting parses</a>, Rutgers Optimality Archive, 2010.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,2).

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%H <a href="/index/Lu#Lucas">Index entries for Lucas sequences.</a>

%F _Wolfdieter Lang_ observes that a(n) = (-I*sqrt(2))^n*U(n, I/sqrt(2)) where U(n, x) is the Chebyshev U-polynomial.

%F G.f.: x/(1 - 2*x - 2*x^2).

%F From _Paul Barry_, Sep 17 2003: (Start)

%F E.g.f. exp(x)*(sinh(sqrt(3)x)/sqrt(3) + cosh(sqrt(3)x)).

%F a(n) = (1 + sqrt(3))^n(1/2 + sqrt(3)/6) + (1 - sqrt(3))^n*(1/2 - sqrt(3)/6).

%F Binomial transform of 1, 1, 3, 3, 9, 9, ... Binomial transform is A079935. (End)

%F a(n) = Sum_{k=0..floor(n/2)} binomial(n - k, k)*2^(n - k). - _Paul Barry_, Jul 13 2004

%F a(n) = A080040(n) - A028860(n+1). - _Creighton Dement_, Jan 19 2005

%F a(n) = Sum_{k=0..n} A112899(n,k). - _Philippe Deléham_, Nov 21 2007

%F a(n) = Sum_{k=0..n} A063967(n,k). - _Philippe Deléham_, Nov 03 2006

%F a(n) = ((1 + sqrt(3))^n - (1 - sqrt(3))^n)/(2*sqrt(3)).

%F a(n) = Sum_{k=0..n} binomial(n, 2*k + 1) * 3^k.

%F Binomial transform of expansion of sinh(sqrt(3)x)/sqrt(3) (0, 1, 0, 3, 0, 9, ...). E.g.f.: exp(x)*sinh(sqrt(3)*x)/sqrt(3). - _Paul Barry_, May 09 2003

%F a(n) = (1/3)*Sum_{k=1..5} sin(Pi*k/2)*sin(2*Pi*k/3)*(1 + 2*cos(Pi*k/6))^n, n >= 1. - _Herbert Kociemba_, Jun 02 2004

%F a(n+1) = ((3 + sqrt(3))*(1 + sqrt(3))^n + (3 - sqrt(3))*(1 - sqrt(3))^n)/6. - Al Hakanson (hawkuu(AT)gmail.com), Jun 29 2009

%F Antidiagonals sums of A081577. - _J. M. Bergot_, Dec 15 2012

%F G.f.: Q(0)*x/2, where Q(k) = 1 + 1/(1 - x*(4*k + 2 + 2*x)/(x*(4*k + 4 + 2*x) + 1/Q(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Aug 30 2013

%F a(n) = 2^(n - 1)*hypergeom([1 - n/2, (1 - n)/2], [1 - n], -2)) for n >= 3. - _Peter Luschny_, Dec 16 2015

%p a:=0:a:=1:for n from 2 to 50 do a[n]:=2*a[n-1]+2*a[n-2]od: seq(a[n], n=0..33); # _Zerinvary Lajos_, Dec 15 2008

%p with(combstruct):ZL0:=S=Prod(Sequence(Prod(a, Sequence(b))), b):ZL1:=Prod(begin_blockP, Z, end_blockP):ZL2:=Prod(begin_blockLR, Z, Sequence(Prod(mu_length, Z), card>=1), end_blockLR): ZL3:=Prod(begin_blockRL, Sequence(Prod(mu_length, Z), card>=1), Z, end_blockRL):Q:=subs([a=Union(ZL2,ZL2,ZL2), b=ZL1], ZL0), begin_blockP=Epsilon, end_blockP=Epsilon, begin_blockLR=Epsilon, end_blockLR=Epsilon, begin_blockRL=Epsilon, end_blockRL=Epsilon, mu_length=Epsilon:temp15:=draw([S, {Q}, unlabelled], size=15):seq(count([S, {Q}, unlabelled], size=n)/3, n=2..31); # _Zerinvary Lajos_, Mar 08 2008

%p a := n -> `if`(n<3, n, 2^(n-1)*hypergeom([1-n/2, (1-n)/2], [1-n], -2));

%p seq(simplify(a(n)), n=0..29); # _Peter Luschny_, Dec 16 2015

%t Expand[Table[((1 + Sqrt)^n - (1 - Sqrt)^n)/(2Sqrt), {n, 0, 30}]] (* _Artur Jasinski_, Dec 10 2006 *)

%t a[n_]:=(MatrixPower[{{1,3},{1,1}},n].{{1},{1}})[[2,1]]; Table[a[n],{n,-1,40}] (* _Vladimir Joseph Stephan Orlovsky_, Feb 19 2010 *)

%t LinearRecurrence[{2, 2}, {0, 1}, 30] (* _Robert G. Wilson v_, Apr 13 2013 *)

%t Round@Table[Fibonacci[n, Sqrt] 2^((n - 1)/2), {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 15 2016 *)

%o (Sage) [lucas_number1(n,2,-2) for n in xrange(0, 30)] # _Zerinvary Lajos_, Apr 22 2009

%o (Sage) # Alternatively:

%o a = BinaryRecurrenceSequence(2,2)

%o print [a(n) for n in (0..29)] # _Peter Luschny_, Aug 29 2016

%o (PARI) Vec(x/(1-2*x-2*x^2)+O(x^99)) \\ _Charles R Greathouse IV_, Jun 10 2011

%o (PARI) A002605(n)=([2,2;1,0]^n)[2,1] \\ _M. F. Hasler_, Aug 06 2018

%o (MAGMA) [Floor(((1 + Sqrt(3))^n - (1 - Sqrt(3))^n)/(2*Sqrt(3))): n in [0..30]]; // _Vincenzo Librandi_, Aug 18 2011

%o a002605 n = a002605_list !! n

%o a002605_list =

%o 0 : 1 : map (* 2) (zipWith (+) a002605_list (tail a002605_list))

%o -- _Reinhard Zumkeller_, Oct 15 2011

%o (MAGMA) [n le 2 select n-1 else 2*Self(n-1) + 2*Self(n-2): n in [1..30]]; // _G. C. Greubel_, Jan 07 2018

%Y First differences are given by A026150.

%Y a(n) = A073387(n, 0), n>=0 (first column of triangle).

%Y Equals (1/3) A083337. First differences of A077846. Pairwise sums of A028860 and abs(A077917).

%Y a(n) = A028860(n)/2 apart from the initial terms.

%Y Row sums of A081577 and row sums of triangle A156710.

%Y The following sequences (and others) belong to the same family: A001333, A000129, A026150, A046717, A015518, A084057, A063727, A002533, A002532, A083098, A083099, A083100, A015519.

%Y Cf. A080953, A052948, A080040, A028859, A030195, A106435, A108898, A125145, A265106, A265107, A265278, A270810, A293005, A293006, A293007.

%Y Cf. A175289 (Pisano periods).

%K nonn,easy

%O 0,3

%A _Colin Mallows_

%E Edited by _N. J. A. Sloane_, Apr 15 2009

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Last modified November 20 02:34 EST 2019. Contains 329323 sequences. (Running on oeis4.)