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A002203 Companion Pell numbers: a(n) = 2a(n-1) + a(n-2).
(Formerly M0360 N0136)
68
2, 2, 6, 14, 34, 82, 198, 478, 1154, 2786, 6726, 16238, 39202, 94642, 228486, 551614, 1331714, 3215042, 7761798, 18738638, 45239074, 109216786, 263672646, 636562078, 1536796802, 3710155682, 8957108166, 21624372014, 52205852194, 126036076402, 304278004998 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Apart from first term, same as A099425. - Peter Shor, May 12 2005

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

REFERENCES

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

Refik Keskin and Olcay Karaatli, Some New Properties of Balancing Numbers and Square Triangular Numbers, Journal of Integer Sequences, Vol. 15 (2012), Article #12.1.4

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.

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

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

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

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

G. L. Alexanderson, Problem B-102, Fib. Quart., 4 (1966), 373.

P. Bhadouria, D. Jhala, B. Singh, Binomial Transforms of the k-Lucas Sequences and its [sic] Properties, The Journal of Mathematics and Computer Science (JMCS), Volume 8, Issue 1, Pages 81-92; sequence L_{2,n}.

Tanya Khovanova, Recursive Sequences

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Eric Weisstein's World of Mathematics, Pell Number.

Wikipedia, Lucas sequence

Index entries for Lucas sequences

Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)

Index entries for sequences related to linear recurrences with constant coefficients

FORMULA

O.g.f.: (2-2x)/(1-2x-x^2). - [Simon Plouffe in his 1992 dissertation]

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

a(n)=A000129(2n)/A000129(n), n>0. - Paul Barry, Feb 06 2004

From Miklos Kristof, Mar 19 2007: (Start)

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

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

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

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)

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)

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)

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)

a(n) = 2*(A000129(n+1)-A000129(n)). - R. J. Mathar, Nov 16 2007

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

G.f.: 3/2 + 1/(2*x) -   3/(1+G(0)), where G(k)= x*(2*k-1) - 1 + 6*x + x*(2*k-1)/G(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Aug 14 2013

MAPLE

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, Mar 08 2008

A002203 := proc(n)

    option remember;

    if n <= 1 then

        2;

    else

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

    end if;

end proc: # R. J. Mathar, May 11 2013

MATHEMATICA

aa = {}; Do[k = Expand[((1 + Sqrt[2])^n + (1 - Sqrt[2])^n)]; AppendTo[aa, k], {n, 0, 30}]; aa [From Artur Jasinski, Dec 23 2008]

a = c = 0; t = {b = 2}; Do[c = a + b + c; AppendTo[t, c]; a = b; b = c, {n, 40}]; t (* or *) LinearRecurrence[{2, 1}, {2, 2}, 40] [From Vladimir Joseph Stephan Orlovsky, Mar 23 2009]

Table[LucasL[n, 2], {n, 0, 30}] [Zerinvary Lajos, Jul 09 2009]

PROG

(Sage) [lucas_number2(n, 2, -1) for n in xrange(0, 29)]# [Zerinvary Lajos, Apr 30 2009]

(Haskell)

a002203 n = a002203_list !! n

a002203_list =

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

-- Reinhard Zumkeller, Oct 03 2011

CROSSREFS

Cf. A000129. a(n) = 2*A001333(n).

Equals A100227(n) + 1.

Bisections are A003499 and A077444.

Sequence in context: A051890 A071109 A005310 * A097341 A142710 A014431

Adjacent sequences:  A002200 A002201 A002202 * A002204 A002205 A002206

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

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

STATUS

approved

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Last modified April 24 09:00 EDT 2014. Contains 240957 sequences.