%I A005248 M0848
%S A005248 2,3,7,18,47,123,322,843,2207,5778,15127,39603,103682,271443,710647,
%T A005248 1860498,4870847,12752043,33385282,87403803,228826127,599074578,
%U A005248 1568397607,4106118243,10749957122,28143753123,73681302247
%N A005248 Bisection of Lucas numbers: a(n) = L(2n) = A000032(2n).
%C A005248 Drop initial 2; then iterates of A050411 do not diverge for these starting 
               values. - David W. Wilson (davidwwilson(AT)comcast.net)
%C A005248 All nonnegative integer solutions of Pell equation a(n)^2 - 5* b(n)^2 
               = +4 together with b(n)=A001906(n), n>=0. W. Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), 
               Aug 31 2004
%C A005248 a(n+1)= B^(n)AB(1), n>=0, with compositions of Wythoff's complementary 
               A(n):=A000201(n) and B(n)=A001950(n) sequences. See the W. Lang link 
               under A135817 for the Wythoff representation of numbers (with A as 
               1 and B as 0 and the argument 1 omitted). E.g. 3=`10`, 7=`010`, 18=`0010`, 
               47=`00010,..., in Wythoff code. a(0)=2= B(1) in Wythoff code.
%C A005248 Contribution from Sarah-Marie Belcastro (smbelcas(AT)toroidalsnark.net), 
               Jul 04 2009: (Start)
%C A005248 Output of Tesler's formula for the number of perfect matchings of an 
               m x n Mobius band where m and n are both even specializes to this 
               sequence for m=2.
%C A005248 Output of Lu and Wu's formula for the number of perfect matchings of 
               an m x n Mobius band where m and n are both even specializes to this 
               sequence for m=2. (End)
%D A005248 T. Crilly, Double sequences of positive integers, Math. Gaz., 69 (1985), 
               263-271.
%D A005248 A. Gougenheim, About the linear sequence of integers such that each term 
               is the sum of the two preceding, Fib. Quart., 9 (1971), 277-295, 
               298.
%D A005248 W. Lu and F. Y. Wu, Close-packed dimers on nonorientable surfaces, Physics 
               Letters A, 293(2002), 235-246. [From Sarah-Marie Belcastro (smbelcas(AT)toroidalsnark.net), 
               Jul 04 2009]
%D A005248 J. O. Shallit, An interesting continued fraction, Math. Mag., 48 (1975), 
               207-211.
%D A005248 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A005248 G. Tesler, Matchings in Graphs on Non-Orientable Surfaces.,Journal of 
               Combinatorial Theory, Series B, 78(2000), 198-231. [From Sarah-Marie 
               Belcastro (smbelcas(AT)toroidalsnark.net), Jul 04 2009]
%H A005248 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A005248 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A005248 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A005248 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PhiNumberSystem.html">Phi Number System</a>
%H A005248 <a href="Sindx_Rea.html#recur1">Index entries for recurrences a(n) = 
               k*a(n - 1) +/- a(n - 2)</a>
%H A005248 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%H A005248 <a href="Sindx_Tu.html#2wis">Index entries for two-way infinite sequences</
               a>
%H A005248 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%F A005248 a(n) = Fibonacci(2*n-1) + Fibonacci(2*n+1).
%F A005248 a(n) = S(n, 3) - S(n-2, 3) = 2*T(n, 3/2) with S(n-1, 3) = A001906(n) 
               and S(-2, x) := -1. U(n, x)=S(n, 2*x) and T(n, x) are Chebyshev's 
               U- and T-polynomials.
%F A005248 a(n) = a(k)*a(n - k) - a(n - 2k) for all k, i.e. a(n) = 2a(n) - a(n) 
               = 3a(n - 1) - a(n - 2) = 7a(n - 2) - a(n - 4) = 18a(n - 3) - a(n 
               - 6) = 47a(n - 4) - a(n - 8) etc. a(2n) = a(n)^2 - 2. - Henry Bottomley 
               (se16(AT)btinternet.com), May 08 2001
%F A005248 a(n) = A060924(n-1, 0) = 3*A001906(n) - 2*A001906(n-1), n >= 1. - Wolfdieter 
               Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Apr 26 2001
%F A005248 a(n) ~ phi^(2*n) where phi=(1+sqrt(5))/2.- Joe Keane (jgk(AT)jgk.org), 
               May 15 2002.
%F A005248 G.f.: (2-3x)/(1-3x+x^2). a(0)=2, a(1)=3, a(n)=3*a(n-1)-a(n-2)=a(-n).
%F A005248 a(n)=phi^(2*n)+phi^(-2*n) where phi=(sqrt(5)+1)/2, the golden ratio. 
               E.g. a(4)=47 because phi^(8) + phi^(-8)=47 - Dennis P. Walsh (dwalsh(AT)mtsu.edu), 
               Jul 24 2003
%F A005248 With interpolated zeros, trace(A^n)/4, where A is the adjacency matrix 
               of path graph P_4. Binomial transform is then A049680. - Paul Barry 
               (pbarry(AT)wit.ie), Apr 24 2004
%F A005248 a(n)={floor((3+sqrt(5))^n) + 1}/2^n. - Lekraj Beedassy (blekraj(AT)yahoo.com), 
               Oct 22 2004
%F A005248 a(n) = ((3-sqrt(5))^n + (3+sqrt(5))^n)*(1/2)^n ( Note: substituting the 
               number 1 for 3 in the last equation gives A000204, substituting 5 
               for 3 gives A020876 ) - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), 
               Apr 19 2005
%F A005248 a(n)=1/(n+1/2)*sum_{k=0...n}B(2k)*L(2n+1-2k)*binomial(2n+1, 2k) where 
               B(2k) is the (2k)-th Bernoulli number. - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Nov 02 2005
%F A005248 a(n) = term (1,1) in the 1x2 matrix [2,3] . [3,1; -1,0]^n. [From Alois 
               P. Heinz (heinz(AT)hs-heilbronn.de), Jul 31 2008]
%F A005248 a(n)=2*cosh(2*n*psi). Psi is ln((1+sqrt5)/2). Offset 0. a(3)=18. [From 
               Al Hakanson (hawkuu(AT)gmail.com), Mar 21 2009]
%F A005248 Contribution from Sarah-Marie Belcastro (smbelcas(AT)toroidalsnark.net), 
               Jul 04 2009: (Start)
%F A005248 (L_{2n}-F_{2n})/2+F_{2n+1} (Tesler)
%F A005248 \prod_{r=1}^{n}(1+4\sin^2(((4r-1)\pi)/(4n))) (Lu/Wu) (End)
%e A005248 2 + 3*x + 7*x^2 + 18*x^3 + 47*x^4 + 123*x^5 + 322*x^6 + 843*x^7 + ... 
               - Michael Somos Aug 11 2009
%p A005248 A005248:=-(-2+3*z)/(1-3*z+z**2); [Conjectured (correctly) by S. Plouffe 
               in his 1992 dissertation.]
%p A005248 (Maple) a := n -> (Matrix([[2,3]]).Matrix([[3,1],[-1,0]])^n)[1,1]; seq 
               (a(n), n=0..26); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), 
               Jul 31 2008]
%t A005248 a[0] = 2; a[1] = 3; a[n_] := 3a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 
               27}] (from Robert G. Wilson v Jan 30 2004)
%t A005248 Fibonacci[1 + 2n] + 1/2 (-Fibonacci[2n] + LucasL[2n]) (Tesler) [From 
               Sarah-Marie Belcastro (smbelcas(AT)toroidalsnark.net), Jul 04 2009]
%o A005248 (PARI) a(n)=fibonacci(2*n+1)+fibonacci(2*n-1)
%o A005248 (PARI) a(n)=2*subst(poltchebi(abs(n)),x,3/2)
%o A005248 sage: [lucas_number2(n,3,1) for n in range(37)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jun 25 2008
%o A005248 (PARI) {a(n) = fibonacci(2*n + 1) + fibonacci(2*n - 1)} - Michael Somos 
               Aug 11 2009
%Y A005248 Cf. A000032, A002878 (odd indexed Lucas numbers), A001906 (Chebyshev 
               S(n-1, 3)), a(n)=sqrt(4+5*A001906(n)^2).
%Y A005248 a(n) = A005592(n)+1 = A004146(n)+2 = A065034(n)-1. First differences 
               of A002878. Pairwise sums of A001519.
%Y A005248 First row of array A103997.
%Y A005248 Sequence in context: A058334 A131093 A002864 this_sequence A032102 A100388 
               A160181
%Y A005248 Adjacent sequences: A005245 A005246 A005247 this_sequence A005249 A005250 
               A005251
%K A005248 nonn,easy
%O A005248 0,1
%A A005248 N. J. A. Sloane (njas(AT)research.att.com).
%E A005248 More terms from James A. Sellers (sellersj(AT)math.psu.edu), May 29 2000
%E A005248 Additional comments from Michael Somos, Jun 23 2001