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A002203
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Companion Pell numbers: a(n) = 2a(n-1) + a(n-2).
(Formerly M0360 N0136)
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65
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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;
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history;
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OFFSET
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0,1
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COMMENTS
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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
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REFERENCES
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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.
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).
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LINKS
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_Reinhard Zumkeller_, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
_Simon 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.
_Simon Plouffe_, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 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
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FORMULA
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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
Comments 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
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MAPLE
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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
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
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MATHEMATICA
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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}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 09 2009]
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PROG
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(Sage) [lucas_number2(n, 2, -1) for n in xrange(0, 29)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 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
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CROSSREFS
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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
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KEYWORD
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nonn,easy,changed
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), Dec 03 2001
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STATUS
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approved
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