%I A002878 M3420 N1384
%S A002878 1,4,11,29,76,199,521,1364,3571,9349,24476,64079,167761,439204,1149851,
%T A002878 3010349,7881196,20633239,54018521,141422324,370248451,969323029,
%U A002878 2537720636,6643838879,17393796001,45537549124,119218851371
%N A002878 Bisection of Lucas sequence: a(n) = L(2n+1).
%C A002878 In any generalized Fibonacci sequence {f(i)}, sum_{i=0..4n+1} f(i) = 
               a(n) f(2n+2). - Lekraj Beedassy (blekraj(AT)yahoo.com), Dec 31 2002
%C A002878 The continued fraction expansion for F((2n+1)*(k+1))/F((2n+1)*k) k>=1 
               is [a(n),a(n),...,a(n)] where there are exactly k elements (F(n) 
               denotes the n-th Fibonacci number). E.g. continued fraction for F(12)/
               F(9) is [4, 4,4]. - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 
               10 2003
%C A002878 See A135064 for a possible connection with Galois groups of quintics.
%C A002878 Sequence of all positive integers k such that continued fraction [k,k,
               k,k,k,k,...] belongs to Q(sqrt(5)). - Thomas Baruchel Sep 15 2003
%C A002878 All positive integer solutions of Pell equation a(n)^2 - 5*b(n)^2 = -4 
               together with b(n)=A001519(n), n>=0.
%C A002878 a(n) = L(n,-3)*(-1)^n, where L is defined as in A108299; see also A001519 
               for L(n,+3).
%C A002878 Inverse binomial transform of A030191 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Oct 04 2005
%C A002878 General recurrence is a(n)=(a(1)-1)*a(n-1)-a(n-2), a(1)>=4, lim n->infinity 
               a(n)= x*(k*x+1)^n, k =(a(1)-3), x=(1+sqrt((a(1)+1)/(a(1)-3)))/2. 
               Examples in OEIS: a(1)=4 gives A002878, primes in it A121534. a(1)=5 
               gives A001834, primes in it A086386. a(1)=6 gives A030221, primes 
               in it not in OEIS {29,139,3191,...}. a(1)=7 gives A002315, primes 
               in it A088165. a(1)=8 gives A033890, primes in it not in OEIS (does 
               there exist any ?). a(1)=9 gives A057080, primes in it not in OEIS 
               {71,34649,16908641,...}. a(1)=10 gives A057081, primes in it not 
               in OEIS {389806471,192097408520951,...}. [From Ctibor O. Zizka (ctibor.zizka(AT)seznam.cz), 
               Sep 02 2008]
%C A002878 Let r = (2n+1), then a(n), n>0 = PRODUCT_{k=1,[(r-1)/2] (1 + Sin^2 k*Pi/
               r); e.g., a(3) = 29 = (3.4450418679...)*(4.801937735...)*(1.753020396...). 
               [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 26 2008]
%C A002878 a(n+1) is the Hankel transform of A001700(n)+A001700(n+1). [From Paul 
               Barry (pbarry(AT)wit.ie), Apr 21 2009]
%D A002878 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002878 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002878 N. D. Cahill, J. D. D'Errico and J. P. Spencer, "Complex Factorizations 
               of the Fibonacci and Lucas Numbers"; Fibonacci Quarterly, 1(41):13-19, 
               2003. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 26 2008]
%D A002878 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.
%H A002878 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A002878 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A002878 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 A002878 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 A002878 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               FibonacciPolynomial.html">Fibonacci Polynomial</a>
%H A002878 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A002878 a(n) ~ phi^(2*n+1) - Joe Keane (jgk(AT)jgk.org), May 15 2002
%F A002878 Let q(n, x)=sum(i=0, n, x^(n-i)*binomial(2*n-i, i)); then (-1)^n*q(n, 
               -1)=a(n) - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 10 2002
%F A002878 a(n) = A005248(n+1) - A005248(n) = sum(A005248:0, n) - 1. - Lekraj Beedassy 
               (blekraj(AT)yahoo.com), Dec 31 2002
%F A002878 a(n) = 2^(-n)*A082762(n) = 4^(-n)*Sum_{k>=0} binomial(2*n+1, 2*k)*5^k; 
               see A091042 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 01 
               2004
%F A002878 a(n)=(-1)^n*sum(k=0, n, (-5)^k*binomial(n+k, n-k)) - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               May 09 2004
%F A002878 Both bisection and binomial transform of A000204. a(n)=Fib(2n)+Fib(2n+2). 
               - Paul Barry (pbarry(AT)wit.ie), May 27 2004
%F A002878 a(n+1)=3*a(n)-a(n-1). G.f.: (1+x)/(1-3*x+x^2). a(n)= S(2*n, sqrt(5)) 
               = S(n, 3)+S(n-1, 3); S(n, x) := U(n, x/2), Chebyshev polynomials 
               of 2nd kind, A049310. S(n, 3)= A001906(n+1) (even indexed Fibonacci 
               numbers).
%F A002878 a(n)=(1/2)*[(3/2)+(1/2)*sqrt(5)]^n+(1/2)*[(3/2)+(1/2)*sqrt(5)]^n*sqrt(5)-(1/
               2)*[(3/2)-(1/2)*sqrt(5)]^n *sqrt(5)+(1/2)*[(3/2)-(1/2)*sqrt(5)]^n, 
               with n>=0 [From Paolo P. Lava (ppl(AT)spl.at), Nov 21 2008]
%F A002878 a(n)=the numerators of sinh((2*n-1)*psi) where the denominators are 2. 
               Psi=ln((1+sqrt5)/2). Offset 1. a(3)=11. [From Al Hakanson (hawkuu(AT)gmail.com), 
               Mar 25 2009]
%p A002878 A002878:=(1+z)/(1-3*z+z**2); [Conjectured (correctly) by S. Plouffe in 
               his 1992 dissertation.]
%t A002878 f[n_] := FullSimplify[GoldenRatio^n - GoldenRatio^-n]; Table[f[n], {n, 
               1, 55, 2}] (* or *)
%t A002878 a[1] = 1; a[2] = 4; a[n_] := a[n] = 3a[n - 1] - a[n - 2]; Array[a, 28] 
               (* or *)
%o A002878 (Other) sage: [(lucas_number2(n,3,1)-lucas_number2(n-1,3,1)) for n in 
               xrange(1, 28)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Nov 10 2009]
%Y A002878 Cf. A000204. a(n)= A060923(n, 0).
%Y A002878 Cf. A005248 [L(2n) = bisection (even n) of Lucas sequence].
%Y A002878 Cf. A001906 [F(2n) = bisection (even n) of Fibonacci sequence].
%Y A002878 Sequence in context: A027968 A027970 A027972 this_sequence A098149 A124861 
               A110579
%Y A002878 Adjacent sequences: A002875 A002876 A002877 this_sequence A002879 A002880 
               A002881
%K A002878 nonn,easy,new
%O A002878 0,2
%A A002878 N. J. A. Sloane (njas(AT)research.att.com).
%E A002878 More terms from James A. Sellers (sellersj(AT)math.psu.edu), May 29 2000
%E A002878 Chebyshev and Pell comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), 
               Aug 31 2004