%I A002203 M0360 N0136
%S A002203 2,2,6,14,34,82,198,478,1154,2786,6726,16238,39202,94642,228486,551614,
%T A002203 1331714,3215042,7761798,18738638,45239074,109216786,263672646,
%U A002203 636562078,1536796802,3710155682,8957108166,21624372014,52205852194
%N A002203 Companion Pell numbers: a(n) = 2a(n-1) + a(n-2).
%C A002203 Apart from first term, same as A099425. - Peter Shor, May 12 2005.
%D A002203 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002203 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002203 S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques 
               Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%D A002203 P. Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 
               1968, vol. 2, p. 76.
%D A002203 E. Lucas, "Th\'eorie des Fonctions Num\'eriques Simplement P\'eriodiques, 
               I", Amer. J. Math., 1 (1878), 184-240. Translated as E. Lucas, The 
               Theory of Simply Periodic Numerical Functions, Fibonacci Association, 
               1969.
%D A002203 Problem B-102, Fib. Quart., 4 (1966), 373.
%D A002203 P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 
               2nd ed., 1989, p. 43.
%H A002203 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A002203 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A002203 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A002203 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A002203 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PellNumber.html">Link to a section of The World of Mathematics.</
               a>
%H A002203 <a href="Sindx_Rea.html#recur1">Index entries for recurrences a(n) = 
               k*a(n - 1) +/- a(n - 2)</a>
%F A002203 O.g.f.: (2-2x)/(1-2x-x^2). - Len Smiley (smiley(AT)math.uaa.alaska.edu), 
               Dec 02 2001
%F A002203 a(n)=(1+sqrt(2))^n+(1-sqrt(2))^n. - Mario Catalani (mario.catalani(AT)unito.it), 
               Mar 17 2003
%F A002203 a(n)=A000129(2n)/A000129(n), n>0. - Paul Barry (pbarry(AT)wit.ie), Feb 
               06 2004
%F A002203 Comments from Miklos Kristof (kristmikl(AT)freemail.hu), Mar 19 2007: 
               (Start)
%F A002203 Let F(n)=A000129=Pell numbers, L(n)=a(n)=Companion Pell numbers:
%F A002203 L(n+m)+(-1)^m*L(n-m)=L(n)*L(m)
%F A002203 L(n+m)-(-1)^m*L(n-m)=8*F(n)*F(m)
%F A002203 L(n+m+k)+(-1)^k*L(n+m-k)+(-1)^m*(L(n-m+k)+(-1)^k*L(n-m-k))=L(n)*L(m)*L(k)
%F A002203 L(n+m+k)-(-1)^k*L(n+m-k)+(-1)^m*(L(n-m+k)-(-1)^k*L(n-m-k))=8*F(n)*L(m)*F(k)
%F A002203 L(n+m+k)+(-1)^k*L(n+m-k)-(-1)^m*(L(n-m+k)+(-1)^k*L(n-m-k))=8*F(n)*F(m)*L(k)
%F A002203 L(n+m+k)-(-1)^k*L(n+m-k)-(-1)^m*(L(n-m+k)-(-1)^k*L(n-m-k))=8*L(n)*F(m)*F(k) 
               (End)
%F A002203 a(n)=2*[A000129(n+1)-A000129(n)]. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Nov 16 2007
%p A002203 A002203:=2*(-1+z)/(-1+2*z+z**2); [Conjectured by S. Plouffe in his 1992 
               dissertation.]
%p A002203 with(combstruct):ZL0:=S=Prod(Sequence(Prod(a, Sequence(b))), a):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(ZL3, ZL3), 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), n=2..30); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Mar 08 2008
%t A002203 aa = {}; Do[k = Expand[((1 + Sqrt[2])^n + (1 - Sqrt[2])^n)]; AppendTo[aa, 
               k], {n, 0, 30}]; aa [From Artur Jasinski (grafix(AT)csl.pl), Dec 
               23 2008]
%t A002203 a=0;b=2;c=0;lst={b};Do[c=a+b+c;AppendTo[lst,c];a=b;b=c,{n,5!}];lst [From 
               Vladimir Orlovsky (4vladimir(AT)gmail.com), Mar 23 2009]
%t A002203 Table[LucasL[n, 2], {n, 0, 30}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jul 09 2009]
%o A002203 (Other) sage: [lucas_number2(n,2,-1) for n in xrange(0, 29)]# [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Apr 30 2009]
%Y A002203 Cf. A000129. a(n) = 2*A001333(n).
%Y A002203 Equals A100227(n) + 1.
%Y A002203 Bisections are A003499 and A077444.
%Y A002203 Sequence in context: A051890 A071109 A005310 this_sequence A097341 A142710 
               A014431
%Y A002203 Adjacent sequences: A002200 A002201 A002202 this_sequence A002204 A002205 
               A002206
%K A002203 nonn,easy
%O A002203 0,1
%A A002203 N. J. A. Sloane (njas(AT)research.att.com).
%E A002203 More terms from Larry Reeves (larryr(AT)acm.org), Dec 03 2001