%I A014551
%S A014551 2,1,5,7,17,31,65,127,257,511,1025,2047,4097,8191,16385,32767,65537,
%T A014551 131071,262145,524287,1048577,2097151,4194305,8388607,16777217,
%U A014551 33554431,67108865,134217727,268435457,536870911,1073741825,2147483647
%N A014551 Jacobsthal-Lucas numbers.
%C A014551 Also gives the number of points of period n in the subshift of finite 
               type corresponding to the square matrix A=[1,2;1,0] (this is then 
               given by trace(A^n)). - Thomas Ward (t.ward(AT)uea.ac.uk), Mar 07 
               2001
%C A014551 Sequence is identical to its signed inverse binomial transform. - Paul 
               Curtz (bpcrtz(AT)free.fr), Jul 11 2008
%C A014551 a(n) can be expressed in terms of values of the Fibonacci polynomials 
               F_n(x), computed at x=1/sqrt(2). [From Tewodros Amdeberhan (tewodros(AT)math.mit.edu), 
               Dec 15 2008]
%D A014551 G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence 
               Sequences, Amer. Math. Soc., 2003; see esp. pp. 180, 255.
%D A014551 Horadam, A. F. ``Jacobsthal and Pell Curves.'' Fib. Quart. 26, 79-83, 
               1988.
%D A014551 Horadam, A. F. ``Jacobsthal Representation Numbers.'' Fib Quart. 34, 
               40-54, 1996.
%D A014551 Lind and Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge 
               University Press, 1995. (General material on subshifts of finite 
               type)
%H A014551 T. D. Noe, <a href="b014551.txt">Table of n, a(n) for n=0..200</a>
%H A014551 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A014551 Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/
               index.html">Arithmetic and growth of periodic orbits</a>, J. Integer 
               Seqs., Vol. 4 (2001), #01.2.1.
%H A014551 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               JacobsthalNumber.html">Link to a section of The World of Mathematics.</
               a>
%H A014551 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%H A014551 T. Amdeberhan, <a href="http://www.math.tulane.edu/~tamdeberhan/fibon.pdf">
               A note on Fibonacci-type polynomials </a> [From Tewodros Amdeberhan 
               (tewodros(AT)math.mit.edu), Dec 15 2008]
%F A014551 a(n+1) = 2 * a(n) - (-1)^n * 3.
%F A014551 a(n) = 2^n + (-1)^n. G.f.: (2-x)/(1-x-2*x^2). - Len Smiley (smiley(AT)math.uaa.alaska.edu), 
               Dec 07 2001
%F A014551 E.g.f.: exp(x)+exp(-2x) produces a signed version. - Paul Barry (pbarry(AT)wit.ie), 
               Apr 27 2003
%F A014551 a(n+1)=Sum{k=0..floor(n/2), binomial(n-1, 2k)3^(2k)/2^(n-2)}. - Paul 
               Barry (pbarry(AT)wit.ie), Feb 21 2003
%F A014551 0, 1, 5, 7 ... is 2^n-2*0^n+(-1)^n, the 2nd inverse binomial transform 
               of (2^n-1)^2 (A060867) - Paul Barry (pbarry(AT)wit.ie), Sep 05 2003
%F A014551 a(n)=2T(n, i/(2sqrt(2)))(-i*sqrt(2))^n with i^2=-1 - Paul Barry (pbarry(AT)wit.ie), 
               Nov 17 2003
%F A014551 a(n)=(A078008(n)+A001045(n+1)) - Paul Barry (pbarry(AT)wit.ie), Feb 12 
               2004
%F A014551 a(n)=2*A001045(n+1)-A001045(n) - Paul Barry (pbarry(AT)wit.ie), Mar 22 
               2004
%F A014551 a(0)=2, a(1)=1, a(n)=a(n-1)+2*a(n-2) for n>1 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Nov 07 2006
%F A014551 a(2n+1)=Product(d divides 2n+1, cyclotomic(d,2)). a(2^k*(2n+1))=Product(d 
               divides 2n+1,cyclotomic(2d,2^(2^k))) - Miklos Kristof (kristmikl(AT)freemail.hu), 
               Mar 12 2007
%F A014551 a(n)=2^{(n-1)/2}F_{n-1}(1/sqrt(2))+2^{(n+2)/2}F_{n-2}(1/sqrt(2)). [From 
               Tewodros Amdeberhan (tewodros(AT)math.mit.edu), Dec 15 2008]
%o A014551 (Other) sage: [lucas_number2(n,1,-2) for n in xrange(0, 32)]# [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Apr 30 2009]
%Y A014551 Cf. A001045 A019322 A066845.
%Y A014551 Sequence in context: A002251 A093545 A005297 this_sequence A088014 A141507 
               A059274
%Y A014551 Adjacent sequences: A014548 A014549 A014550 this_sequence A014552 A014553 
               A014554
%K A014551 nonn,nice
%O A014551 0,1
%A A014551 Eric Weisstein (eric(AT)weisstein.com)
%E A014551 More terms from Patrick De Geest (pdg(AT)worldofnumbers.com), Jun 15 
               1998.