%I A002605
%S A002605 0,1,2,6,16,44,120,328,896,2448,6688,18272,49920,136384,372608,1017984,
%T A002605 2781184,7598336,20759040,56714752,154947584,423324672,1156544512,
%U A002605 3159738368,8632565760,23584608256,64434348032,176037912576,480944521216,
               1313964867584
%N A002605 a(n)=2(a(n-1)+a(n-2)), a(0)=0, a(1)=1.
%C A002605 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 A002605 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 (kociemba(AT)t-online.de), Jun 02 2004
%C A002605 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 (hillcino368(AT)gmail.com), 
               Sep 25 2005
%C A002605 The Hankel transform of this sequence is [1,2,0,0,0,0,0,0,0,...]. - Philippe 
               DELEHAM (kolotoko(AT)wanadoo.fr), Nov 21 2007
%C A002605 [1,3; 1,1]^n *[1,0] = [A026150(n), A002605(n)]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Mar 21 2008
%C A002605 (1+sqrt(3))^n = A026150(n) + A002605(n)*sqrt(3) - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Mar 21 2008
%C A002605 a(n) is the number of ways to tile a board of length n using red and 
               blue tiles of length one and two. [From Geoffrey Critzer (critzer.geoffrey(AT)usd443), 
               Feb 07 2009]
%C A002605 Starting with offset 1 = INVERT transform of the Jacobsthal sequence, 
               A001045: (1, 1, 3, 5, 11, 21,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               May 12 2009]
%D A002605 John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see 
               p. 16.
%D A002605 A. F. Horadam, Special properties of the sequence w_n(a,b; p,q), Fib. 
               Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=q=2.
%D A002605 W. Lang, On polynomials related to powers of the generating function 
               of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Eqs. (39), 
               (41) and (45), lhs, m=2.
%H A002605 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=476">
               Encyclopedia of Combinatorial Structures 476</a>
%H A002605 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A002605 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A002605 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A002605 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) observes 
               that a(n)=(-I*sqrt(2))^n*U(n, I/sqrt(2)), U(n, x) = Chebyshev U-polynomial.
%F A002605 G.f.: 1/(1-2*x-2*x^2).
%F A002605 E.g.f. exp(x)(sinh(sqrt(3)x)/sqrt(3)+cosh(sqrt(3)x)); a(n)=(1+sqrt(3))^n(1/
               2+sqrt(3)/6)+(1-sqrt(3))^n(1/2-sqrt(3)/6). Binomial transform of 
               1, 1, 3, 3, 9, 9, ... Binomial transform is A079935. - Paul Barry 
               (pbarry(AT)wit.ie), Sep 17 2003
%F A002605 a(n)=(1/3)*Sum(k, 1, 5, Sin(Pi*k/2)Sin(Pi*k/3)(1+2Cos(Pi*k/6))^(n+1)) 
               - Herbert Kociemba (kociemba(AT)t-online.de), Jun 02 2004
%F A002605 a(n)= sum{k=0..floor(n/2), binomial(n-k, k)2^(n-k)} - Paul Barry (pbarry(AT)wit.ie), 
               Jul 13 2004
%F A002605 A002605(n) = A080040(n) - A028860(n+1) - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), 
               Jan 19 2005
%F A002605 a(n)=Sum{k, 0<=k<=n}A112899(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Nov 21 2007
%F A002605 a(n)=Sum_{k, 0<=k<=n}A063967(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Nov 03 2006
%F A002605 a(n)=((1 + sqrt(3))^n - (1 - sqrt(3))^n)/(2*sqrt(3)); a(n)=Sum{k=0..n, 
               binomial(n, 2k+1)3^k}; G.f.: x/(1-2x-2x^2)
%F A002605 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 (pbarry(AT)wit.ie), 
               May 09 2003
%F A002605 a(n)=(1/3)*Sum(k, 1, 5, Sin(Pi*k/2)Sin(2Pi*k/3)(1+2Cos(Pi*k/6))^n) - 
               Herbert Kociemba (kociemba(AT)t-online.de), Jun 02 2004
%F A002605 a(n)=((3+sqrt3)(1+sqrt3)^n+(3-sqrt3)(1-sqrt3)^n)/6 offset 0. Add a leading 
               0. [From Al Hakanson (hawkuu(AT)gmail.com), Jun 29 2009]
%p A002605 a[0]:=0:a[1]:=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);# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 
               15 2008]
%p A002605 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 (zerinvarylajos(AT)yahoo.com), 
               Mar 08 2008
%t A002605 Expand[Table[((1 + Sqrt[3])^n - (1 - Sqrt[3])^n)/(2Sqrt[3]), {n, 0, 30}]] 
               - Artur Jasinski (grafix(AT)csl.pl), Dec 10 2006
%o A002605 (Other) sage: [lucas_number1(n,2,-2) for n in xrange(0, 30)] # [From 
               Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]
%Y A002605 First differences are given by A026150.
%Y A002605 a(n) = A073387(n, 0), n>=0 (first column of triangle).
%Y A002605 Cf. A080953, A026150, A052948, A077846, A080040.
%Y A002605 Cf. A002605, A028859.
%Y A002605 Equals (1/3) A083337. First differences of A077846. Pairwise sums of 
               A028860 and |A077917|.
%Y A002605 a(n)=A028860(n)/2 apart from the initial terms. [From Philippe DELEHAM 
               (kolotoko(AT)wanadoo.fr), Nov 19 2008]
%Y A002605 Row sums of Pascal-(1,2,1) triangle A081577. - Paul Barry (pbarry(AT)wit.ie), 
               Jan 24 2005
%Y A002605 Equals row sums of triangle A156710 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Feb 14 2009]
%Y A002605 The following sequences (and others) belong to the same family: A001333, 
               A000129, A026150, A002605, A046717, A015518, A084057, A063727, A002533, 
               A002532, A083098, A083099, A083100, A015519.
%Y A002605 Sequence in context: A027068 A118041 A105073 this_sequence A026134 A105696 
               A074413
%Y A002605 Adjacent sequences: A002602 A002603 A002604 this_sequence A002606 A002607 
               A002608
%K A002605 nonn
%O A002605 0,3
%A A002605 C. L. Mallows (colinm(AT)research.avayalabs.com)
%E A002605 Edited by N. J. A. Sloane, Apr 15 2009