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A002878 Bisection of Lucas sequence: a(n) = L(2*n+1).
(Formerly M3420 N1384)
75
1, 4, 11, 29, 76, 199, 521, 1364, 3571, 9349, 24476, 64079, 167761, 439204, 1149851, 3010349, 7881196, 20633239, 54018521, 141422324, 370248451, 969323029, 2537720636, 6643838879, 17393796001, 45537549124, 119218851371, 312119004989, 817138163596, 2139295485799 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

In any generalized Fibonacci sequence {f(i)}, sum_{i=0..4n+1} f(i) = a(n)*f(2n+2). - Lekraj Beedassy, Dec 31 2002

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, Apr 10 2003

See A135064 for a possible connection with Galois groups of quintics.

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

All positive integer solutions of Pell equation a(n)^2 - 5*b(n)^2 = -4 together with b(n)=A001519(n), n>=0.

a(n) = L(n,-3)*(-1)^n, where L is defined as in A108299; see also A001519 for L(n,+3).

Inverse binomial transform of A030191. - Philippe Deléham, Oct 04 2005

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,...}. - Ctibor O. Zizka, Sep 02 2008

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...). - Gary W. Adamson, Nov 26 2008

a(n+1) is the Hankel transform of A001700(n)+A001700(n+1). - Paul Barry, Apr 21 2009

a(n) is equal to the permanent of the (2n) X (2n) tridiagonal matrix with sqrt(5)'s along the main diagonal, i's along the superdiagonal and the subdiagonal (i is the imaginary unit), and 0's everywhere else. - John M. Campbell, Jun 09 2011

Conjecture: for n>0, a(n)=sqrt(Fibonacci(4n+3)+sum_{k=2..2n}Fibonacci(2k)). - Alex Ratushnyak, May 06 2012

Pisano period lengths: 1, 3, 4, 3, 2, 12, 8, 6, 12, 6, 5, 12, 14, 24, 4, 12, 18, 12, 9, 6,... . - R. J. Mathar, Aug 10 2012

The continued fraction [a(n);a(n),a(n),...] = phi^(2n+1), where phi is the golden ratio, A001622. - Thomas Ordowski, Jun 05 2013

Solutions (x, y) = (a(n), a(n+1)) satisfying  x^2 + y^2 = 3xy + 5. - Michel Lagneau, Feb 01 2014

REFERENCES

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).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

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.

M. Elder and A. Kalka, Logspace computations for rigid Garside groups, arXiv preprint arXiv:1310.0933, 2013

A. Gougenheim, About the linear sequence of integers such that each term is the sum of the two preceding Part 1 Part 2, Fib. Quart., 9 (1971), 277-295, 298.

Tanya Khovanova, Recursive Sequences

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Eric Weisstein's World of Mathematics, Fibonacci Polynomial

Index entries for sequences related to Chebyshev polynomials.

Index entries for sequences related to linear recurrences with constant coefficients, signature (3,-1).

FORMULA

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).

a(n) ~ phi^(2*n+1). - Joe Keane (jgk(AT)jgk.org), May 15 2002

a(n) = phi^(2*n+1)-phi^(-2*n-1), n >=0. - Paolo P. Lava, Jan 03 2011

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, Nov 10 2002

a(n) = A005248(n+1) - A005248(n) = sum(A005248:0, n) - 1. - Lekraj Beedassy, Dec 31 2002

a(n) = 2^(-n)*A082762(n) = 4^(-n)*Sum_{k>=0} binomial(2*n+1, 2*k)*5^k; see A091042. - Philippe Deléham, Mar 01 2004

a(n) = (-1)^n*sum(k=0, n, (-5)^k*binomial(n+k, n-k)). - Benoit Cloitre, May 09 2004

Both bisection and binomial transform of A000204. a(n) = Fib(2n)+Fib(2n+2). - Paul Barry, May 27 2004

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. - Paolo P. Lava, Nov 21 2008

a(n) = the numerators of sinh((2*n-1)*psi) where the denominators are 2. Psi=log((1+sqrt5)/2). Offset 1. a(3)=11. - Al Hakanson (hawkuu(AT)gmail.com), Mar 25 2009

a(n) = A001906(n) + A001906(n+1). - Reinhard Zumkeller, Jan 11 2012

a(n) = floor(phi^(2n+1)), where phi is the golden ratio, A001622. - Thomas Ordowski, Jun 10 2012

a(n) = A014217(2*n+1) = A014217(2*n+2)-A014217(2*n). - Paul Curtz, Jun 11 2013

sum {n >= 0} 1/( a(n) + 5/a(n) ) = 1/2. Compare with A005248, A001906, A075796. - Peter Bala, Nov 29 2013

MAPLE

A002878:=(1+z)/(1-3*z+z**2); # Conjectured (correctly) by Simon Plouffe in his 1992 dissertation.

MATHEMATICA

a[n_] := FullSimplify[GoldenRatio^n - GoldenRatio^-n]; Table[a[n], {n, 1, 55, 2}] (* or *)

a[1] = 1; a[2] = 4; a[n_] := a[n] = 3a[n - 1] - a[n - 2]; Array[a, 28]

PROG

(MAGMA) [ Lucas(2*n +1): n in [0..120]]; // Vincenzo Librandi, Apr 16 2011

(PARI) a(n)=fibonacci(2*n)+fibonacci(2*n+2) \\ Charles R Greathouse IV, Jun 16 2011

(Haskell)

a002878 n = a002878_list !! n

a002878_list = zipWith (+) (tail a001906_list) a001906_list

-- Reinhard Zumkeller, Jan 11 2012

CROSSREFS

Cf. A000204. a(n) = A060923(n, 0), a(n)^2 = A081071(n).

Cf. A005248 [L(2n) = bisection (even n) of Lucas sequence].

Cf. A001906 [F(2n) = bisection (even n) of Fibonacci sequence].

Sequence in context: A027970 A027972 A098149 * A124861 A110579 A024829

Adjacent sequences:  A002875 A002876 A002877 * A002879 A002880 A002881

KEYWORD

nonn,easy,changed

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from James A. Sellers, May 29 2000

Chebyshev and Pell comments from Wolfdieter Lang, Aug 31 2004

STATUS

approved

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Last modified April 20 10:00 EDT 2014. Contains 240779 sequences.