0,1

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

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

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

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.

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

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.

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

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

Index entries for sequences related to linear recurrences with constant coefficients

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.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

Tanya Khovanova, Recursive Sequences

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

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

O.g.f.: (2-2x)/(1-2x-x^2). - Len Smiley (smiley(AT)math.uaa.alaska.edu), Dec 02 2001

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 (pbarry(AT)wit.ie), Feb 06 2004

Comments from Miklos Kristof (kristmikl(AT)freemail.hu), 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 (mathar(AT)strw.leidenuniv.nl), Nov 16 2007

A002203:=2*(-1+z)/(-1+2*z+z**2); [Conjectured by S. Plouffe in his 1992 dissertation.]

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

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]

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]

Table[LucasL[n, 2], {n, 0, 30}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 09 2009]

(Other) sage: [lucas_number2(n, 2, -1) for n in xrange(0, 29)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 30 2009]

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

Equals A100227(n) + 1.

Bisections are A003499 and A077444.

Sequence in context: A051890 A071109 A005310 this_sequence A097341 A142710 A014431

Adjacent sequences: A002200 A002201 A002202 this_sequence A002204 A002205 A002206

nonn,easy

N. J. A. Sloane (njas(AT)research.att.com).

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