login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A005248 Bisection of Lucas numbers: a(n) = L(2*n) = A000032(2*n).
(Formerly M0848 N1067)
79
2, 3, 7, 18, 47, 123, 322, 843, 2207, 5778, 15127, 39603, 103682, 271443, 710647, 1860498, 4870847, 12752043, 33385282, 87403803, 228826127, 599074578, 1568397607, 4106118243, 10749957122, 28143753123, 73681302247, 192900153618, 505019158607, 1322157322203 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Drop initial 2; then iterates of A050411 do not diverge for these starting values. - David W. Wilson

All nonnegative integer solutions of Pell equation a(n)^2 - 5* b(n)^2 = +4 together with b(n)=A001906(n), n>=0. - Wolfdieter Lang, Aug 31 2004

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.

From Sarah-Marie Belcastro, Jul 04 2009: (Start)

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.

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)

Numbers having two 1's in their base-phi representation. - Robert G. Wilson v, Sep 13 2010

The Hosoya index H(n) of the n-cycle graph C_n is given by H(2n-1) = 0 and H(2n) = a(n). - _Eric Weisstein_, Jul 11 2001

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

From Wolfdieter Lang, Feb 18 2013: (Start)

a(n) is also one half of the total number of round trips, each of length 2*n, on the graph P_4 (o-o-o-o) (the simple path with 4 points (vertices) and 3 lines (or edges)). See the array and triangle A198632 for the general case for the graph P_N (there N is n and the length is l=2*k).

O.g.f. for w(4,l) (with zeros for odd l): y*diff(S(4,y),y)/S(4,y) with y=1/x and Chebyshev S-polynomials (coefficients A049310). See A198632, also for a rewritten form. One half of this o.g.f. for x -> sqrt(x) produces the G.f. (2-3x)/(1-3x+x^2) given below. (End)

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

Except for the first term, positive values of x (or y) satisfying x^2 - 7xy + y^2 + 45 = 0. - Colin Barker, Feb 16 2014

Except for the first term, positive values of x (or y) satisfying x^2 - 18xy + y^2 + 320 = 0. - Colin Barker, Feb 16 2014

a(n) are the numbers such that a(n)^2- 2 are Lucas numbers. - Michel Lagneau, Jul 22 2014

REFERENCES

Jeffrey Shallit, An interesting continued fraction, Math. Mag., 48 (1975), 207-211.

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).

G. Tesler, Matchings in Graphs on Non-Orientable Surfaces.,Journal of Combinatorial Theory, Series B, 78 (2000), 198-231

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

P. Bala, Some simple continued fraction expansions for an infinite product, Part 1.

P. Bhadouria, D. Jhala, B. Singh, Binomial Transforms of the k-Lucas Sequences and its [sic] Properties, The Journal of Mathematics and Computer Science (JMCS), Volume 8, Issue 1, Pages 81-92; sequences B_1, T_1 and R_1.

T. Crilly, Double sequences of positive integers, Math. Gaz., 69 (1985), 263-271.

Dale Gerdemann, Collision of Digits "Also interesting are the two bisections of the Lucas numbers A005248 (digit minimizer) and A002878 (digit maximizer). I particularly like the multiples of A005248 because I have this image of the two digits piling up on top of each other and then spreading out like waves".

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

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]

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.

P. J. Szablowski, On moments of Cantor and related distributions, arXiv preprint arXiv:1403.0386, 2014

Eric Weisstein's World of Mathematics, Cycle Graph

Eric Weisstein's World of Mathematics, Hosoya Index

Eric Weisstein's World of Mathematics, Phi Number System

Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)

Index entries for sequences related to Chebyshev polynomials.

Index entries for two-way infinite sequences

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

FORMULA

a(n) = Fibonacci(2*n-1) + Fibonacci(2*n+1).

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.

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, May 08 2001

a(n) = A060924(n-1, 0) = 3*A001906(n) - 2*A001906(n-1), n >= 1. - Wolfdieter Lang, Apr 26 2001

a(n) ~ phi^(2*n) where phi=(1+sqrt(5))/2. - Joe Keane (jgk(AT)jgk.org), May 15 2002.

G.f.: (2 - 3*x) / (1 - 3*x + x^2). a(0)=2, a(1)=3, a(n) = 3*a(n-1) - a(n-2) = a(-n). - Michael Somos, Jun 28 2003

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, Jul 24 2003

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, Apr 24 2004

a(n) = (floor((3+sqrt(5))^n) + 1)/2^n. - Lekraj Beedassy, Oct 22 2004

a(n) = ((3-sqrt(5))^n + (3+sqrt(5))^n)/2^n (Note: substituting the number 1 for 3 in the last equation gives A000204, substituting 5 for 3 gives A020876). - Creighton Dement, Apr 19 2005

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, Nov 02 2005

a(n) = term (1,1) in the 1x2 matrix [2,3] . [3,1; -1,0]^n. - Alois P. Heinz, Jul 31 2008

a(n) = 2*cosh(2*n*psi). Psi is ln((1+sqrt5)/2). Offset 0. a(3)=18. - Al Hakanson, Mar 21 2009

From Sarah-Marie Belcastro, Jul 04 2009: (Start)

a(n)-(a(n)-F(2n))/2-F(2n+1) = 0. (Tesler)

Prod_(r=1)^n (1+4*(sin((4r-1)*pi/(4n)))^2). (Lu/Wu) (End)

a(n) = Fibonacci(2n+6) mod Fibonacci(2n+2), n>1. - Gary Detlefs, Nov 22 2010

a(n) = 5*Fibonacci(n)^2 + 2*(-1)^n. - Gary Detlefs, Nov 22 2010

a(n) = Fibonacci(4*n)/Fibonacci(2*n), n>0. - Gary Detlefs, Dec 26 2010

a(n+2) = 3*a(n+1) - a(n), a(0) = 2, a(1) = 3. - Sture Sjöstedt, Nov 04 2011

a(n) = 2^(2*n)*sum(k=1,2 (cos(k*Pi/5))^(2*n)). - L. Edson Jeffery, Jan 21 2012

From Peter Bala, Jan 04 2013: (Start)

Let F(x) = product {n = 0..inf} (1 + x^(4*n+1))/(1 + x^(4*n+3)). Let alpha = 1/2*(3 - sqrt(5)). This sequence gives the simple continued fraction expansion of 1 + F(alpha) = 2.31829 56058 81914 31334 ... = 2 + 1/(3 + 1/(7 + 1/(18 + ...))).

Also F(-alpha) = 0.64985 97768 07374 32950 has the continued fraction representation 1 - 1/(3 - 1/(7 - 1/(18 - ...))) and the simple continued fraction expansion 1/(1 + 1/((3-2) + 1/(1 + 1/((7-2) + 1/(1 + 1/((18-2) + 1/(1 + ...))))))).

F(alpha)*F(-alpha) has the simple continued fraction expansion 1/(1 + 1/((3^2-4) + 1/(1 + 1/((7^2-4) + 1/(1 + 1/((18^2-4) + 1/(1 + ...))))))).

(End)

G.f.: (W(0) +6)/(5*x) , where W(k) = 5*x*k + x - 6 + 6*x*(5*k-9)/W(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Aug 19 2013

sum {n >= 1} 1/( a(n) - 5/a(n) ) = 1. Compare with A001906, A002878 and A023039. - Peter Bala, Nov 29 2013

a(n) = 2^(n*x) + 2^(-n*x), where x = log(1/2*(sqrt(5) + 3)) / log(2). - Scott A. Brown, Jan 26 2014

0 = a(n) * a(n+2) - a(n+1)^2 - 5 for all n in Z. - Michael Somos, Aug 24 2014

EXAMPLE

G.f. = 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

MAPLE

A005248:=-(-2+3*z)/(1-3*z+z**2); # Conjectured (correctly) by Simon Plouffe in his 1992 dissertation.

a:= n -> (Matrix([[2, 3]]). Matrix([[3, 1], [-1, 0]])^n)[1, 1]: seq (a(n), n=0..30); # Alois P. Heinz, Jul 31 2008

with(combinat):seq(5*fibonacci(n)^2+2*(-1)^n, n= 0..26)

MATHEMATICA

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 *)

Fibonacci[1 + 2n] + 1/2 (-Fibonacci[2n] + LucasL[2n]) (* Tesler. From Sarah-Marie Belcastro, Jul 04 2009 *)

LinearRecurrence[{3, -1}, {2, 3}, 50] (* From Sture Sjöstedt Nov 27 2011 *)

LucasL[Range[0, 60, 2]] (* Harvey P. Dale, Sep 30 2014 *)

PROG

(PARI) {a(n) = fibonacci(2*n + 1) + fibonacci(2*n - 1)}; /* Michael Somos, Jun 23 2002 */

(PARI) {a(n) = 2 * subst( poltchebi( abs(n)), x, 3/2)}; /* Michael Somos, Jun 28 2003 */

(Sage) [lucas_number2(n, 3, 1) for n in range(37)] # Zerinvary Lajos, Jun 25 2008

(MAGMA) [ Lucas(2*n) : n in [0..100]]; // Vincenzo Librandi, Apr 14 2011

(Haskell)

a005248 n = a005248_list !! n

a005248_list = zipWith (+) (tail a001519_list) a001519_list

-- Reinhard Zumkeller, Jan 11 2012

CROSSREFS

Cf. A000032, A002878 (odd indexed Lucas numbers), A001906 (Chebyshev S(n-1, 3)), a(n)=sqrt(4+5*A001906(n)^2).

a(n) = A005592(n)+1 = A004146(n)+2 = A065034(n)-1. First differences of A002878. Pairwise sums of A001519.

First row of array A103997.

Cf. A201157.

Sequence in context: A058334 A131093 A002864 * A032102 A100388 A186232

Adjacent sequences:  A005245 A005246 A005247 * A005249 A005250 A005251

KEYWORD

nonn,easy,changed

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from James A. Sellers, May 29 2000

Additional comments from Michael Somos, Jun 23 2001

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified October 23 13:13 EDT 2014. Contains 248464 sequences.