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A002203 Companion Pell numbers: a(n) = 2a(n-1) + a(n-2), a(0) = a(1) = 2.
(Formerly M0360 N0136)
93

%I M0360 N0136

%S 2,2,6,14,34,82,198,478,1154,2786,6726,16238,39202,94642,228486,

%T 551614,1331714,3215042,7761798,18738638,45239074,109216786,263672646,

%U 636562078,1536796802,3710155682,8957108166,21624372014,52205852194,126036076402,304278004998

%N Companion Pell numbers: a(n) = 2a(n-1) + a(n-2), a(0) = a(1) = 2.

%C Also the number of matchings (independent edge sets) of the n-sunlet graph. - _Eric W. Weisstein_, Mar 09 2016

%C Apart from first term, same as A099425. - _Peter Shor_, May 12 2005

%C The signed sequence 2, -2, 6, -14, 34, -82, 198, -478, 1154, -2786, ... is the Lucas V(-2,-1) sequence. - _R. J. Mathar_, Jan 08 2013

%D P. Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 1968, vol. 2, p. 76.

%D E. Lucas, "Théorie des Fonctions Numériques Simplement Périodiques, I", Amer. J. Math., 1 (1878), 184-240. Translated as E. Lucas, The Theory of Simply Periodic Numerical Functions, Fibonacci Association, 1969.

%D P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 43.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Reinhard Zumkeller, <a href="/A002203/b002203.txt">Table of n, a(n) for n = 0..1000</a>

%H G. L. Alexanderson, <a href="http://www.fq.math.ca/Scanned/4-4/elementary4-4.pdf">Problem B-102</a>, Fib. Quart., 4 (1966), 373.

%H P. Bhadouria, D. Jhala, B. Singh, <a href="http://www.tjmcs.com/includes/files/articles/Vol8_Iss1_81 - 92_Binomial_Transforms_of_the_k-Lucas.pdf">Binomial Transforms of the k-Lucas Sequences and its [sic] Properties</a>, The Journal of Mathematics and Computer Science (JMCS), Volume 8, Issue 1, Pages 81-92; sequence L_{2,n}.

%H S. Falcon, <a href="http://saspublisher.com/wp-content/uploads/2014/06/SJET24C669-675.pdf">On The Generating Functions of the Powers of the K-Fibonacci Numbers</a>, Scholars Journal of Engineering and Technology (SJET), 2014; 2 (4C):669-675.

%H Bakir Farhi, <a href="https://www.emis.de/journals/JIS/VOL22/Farhi/farhi19.html">Summation of Certain Infinite Lucas-Related Series</a>, J. Int. Seq., Vol. 22 (2019), Article 19.1.6.

%H Bernadette Faye, Florian Luca, <a href="http://arxiv.org/abs/1508.05714">Pell Numbers whose Euler Function is a Pell Number</a>, arXiv:1508.05714 [math.NT], 2015.

%H M. C. Firengiz, A. Dil, <a href="http://www.nntdm.net/papers/nntdm-20/NNTDM-20-4-21-32.pdf">Generalized Euler-Seidel method for second order recurrence relations</a>, Notes on Number Theory and Discrete Mathematics, Vol. 20, 2014, No. 4, 21-32.

%H Refik Keskin and Olcay Karaatli, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Karaatli/karaatli5.html">Some New Properties of Balancing Numbers and Square Triangular Numbers</a>, Journal of Integer Sequences, Vol. 15 (2012), Article #12.1.4

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

%H E. Lucas, <a href="http://edouardlucas.free.fr/oeuvres/Theorie_des_fonctions_simplement_periodiques.pdf">Théorie des Fonctions Numériques Simplement Périodiques</a>, I", Amer. J. Math., 1 (1878), 184-240.

%H Edouard Lucas, <a href="http://www.mathstat.dal.ca/FQ/Books/Complete/simply-periodic.pdf">The Theory of Simply Periodic Numerical Functions</a>, Fibonacci Association, 1969. English translation of article "Théorie des Fonctions Numériques Simplement Périodiques, I", Amer. J. Math., 1 (1878), 184-240.

%H Arzu Özkoç, <a href="http://link.springer.com/article/10.1186/s13662-015-0486-7/fulltext.html">Some algebraic identities on quadra Fibona-Pell integer sequence</a>, Advances in Difference Equations, 2015, 2015:148.

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992.

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">1031 Generating Functions and Conjectures</a>, Université du Québec à Montréal, 1992.

%H A. Tekcan, M. Tayat, M. E. Ozbek, <a href="http://dx.doi.org/10.1155/2014/897834">The diophantine equation 8x^2-y^2+8x(1+t)+(2t+1)^2=0 and t-balancing numbers</a>, ISRN Combinatorics, Volume 2014, Article ID 897834, 5 pages.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IndependentEdgeSet.html">Independent Edge Set</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Matching.html">Matching</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PellNumber.html">Pell Number</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SunletGraph.html">Sunlet Graph</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Lucas_sequence#Specific_names">Lucas sequence</a>

%H Abdelmoumène Zekiri, Farid Bencherif, Rachid Boumahdi, <a href="https://www.emis.de/journals/JIS/VOL21/Zekiri/zekiri4.html">Generalization of an Identity of Apostol</a>, J. Int. Seq., Vol. 21 (2018), Article 18.5.1.

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

%H <a href="/index/Rea#recur1">Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,1).

%F a(n) = 2 * A001333(n).

%F a(n) = A100227(n) + 1.

%F O.g.f.: (2-2x)/(1-2x-x^2). - _Simon Plouffe_ in his 1992 dissertation

%F a(n) = (1+sqrt(2))^n+(1-sqrt(2))^n. - Mario Catalani (mario.catalani(AT)unito.it), Mar 17 2003

%F a(n) = A000129(2n)/A000129(n), n>0. - _Paul Barry_, Feb 06 2004

%F From _Miklos Kristof_, Mar 19 2007: (Start)

%F Let F(n)=A000129=Pell numbers, L(n)=a(n)=Companion Pell numbers:

%F L(n+m)+(-1)^m*L(n-m) = L(n)*L(m)

%F L(n+m)-(-1)^m*L(n-m) = 8*F(n)*F(m)

%F 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 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 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 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 a(n) = 2*(A000129(n+1)-A000129(n)). - _R. J. Mathar_, Nov 16 2007

%F G.f.: G(0), where G(k)= 1 + 1/(1 - x*(2*k-1)/(x*(2*k+1) - 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 19 2013

%F a(n) = [x^n] ( (1 + 2*x + sqrt(1 + 4*x + 8*x^2))/2 )^n for n >= 1. - _Peter Bala_, Jun 23 2015

%p A002203 := proc(n)

%p option remember;

%p if n <= 1 then

%p 2;

%p else

%p 2*procname(n-1)+procname(n-2) ;

%p end if;

%p end proc: # _R. J. Mathar_, May 11 2013

%p # second Maple program:

%p a:= n-> (<<0|1>, <1|2>>^n. <<2, 2>>)[1, 1]:

%p seq(a(n), n=0..30); # _Alois P. Heinz_, Jan 26 2018

%t Table[LucasL[n, 2], {n, 0, 30}] (* _Zerinvary Lajos_, Jul 09 2009 *)

%t LinearRecurrence[{2, 1}, {2, 2}, 50] (* _Vincenzo Librandi_, Aug 15 2015 *)

%t Table[(1 - Sqrt[2])^n + (1 + Sqrt[2])^n, {n, 0, 20}] // Expand (* _Eric W. Weisstein_, Oct 03 2017 *)

%t LucasL[Range[0, 20], 2] (* _Eric W. Weisstein_, Oct 03 2017 *)

%t CoefficientList[Series[(2 (1 - x))/(1 - 2 x - x^2), {x, 0, 20}], x] (* _Eric W. Weisstein_, Oct 03 2017 *)

%o (Sage) [lucas_number2(n,2,-1) for n in xrange(0, 29)] # _Zerinvary Lajos_, Apr 30 2009

%o (Haskell)

%o a002203 n = a002203_list !! n

%o a002203_list =

%o 2 : 2 : zipWith (+) (map (* 2) $ tail a002203_list) a002203_list

%o -- _Reinhard Zumkeller_, Oct 03 2011

%o (MAGMA) I:=[2,2]; [n le 2 select I[n] else 2*Self(n-1)+Self(n-2): n in [1..35]] // _Vincenzo Librandi_, Aug 15 2015

%o (PARI) first(m)=my(v=vector(m));v[1]=2;v[2]=2;for(i=3,m,v[i]=2*v[i-1]+v[i-2]);v; \\ _Anders Hellström_, Aug 15 2015

%Y Cf. A000129, A001333, A100227.

%Y Bisections are A003499 and A077444.

%K nonn,easy

%O 0,1

%A _N. J. A. Sloane_

%E More terms from Larry Reeves (larryr(AT)acm.org), Dec 03 2001

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Last modified November 15 01:07 EST 2019. Contains 329142 sequences. (Running on oeis4.)