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A005248
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Bisection of Lucas numbers: a(n) = L(2*n) = A000032(2*n).
(Formerly M0848)
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53
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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; internal format)
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OFFSET
| 0,1
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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.
Contribution 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. [From 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). [From Eric Weisstein, Jul 11 2001]
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REFERENCES
| 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.
J. O. Shallit, An interesting continued fraction, Math. Mag., 48 (1975), 207-211.
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. [From Sarah-Marie Belcastro (smbelcas(AT)toroidalsnark.net), Jul 04 2009]
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LINKS
| T. Crilly, Double sequences of positive integers, Math. Gaz., 69 (1985), 263-271.
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]
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
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).
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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.: (2-3x)/(1-3x+x^2). a(0)=2, a(1)=3, a(n)=3*a(n-1)-a(n-2)=a(-n).
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
Contribution 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
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EXAMPLE
| 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
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MAPLE
| A005248:=-(-2+3*z)/(1-3*z+z**2); # Conjectured (correctly) by S. Plouffe in his 1992 dissertation.
(Maple) a := n -> (Matrix([[2, 3]]).Matrix([[3, 1], [-1, 0]])^n)[1, 1]; seq (a(n), n=0..26); # From Alois P. Heinz, Jul 31 2008
with(combinat):seq(5*fibonacci(n)^2+2*(-1)^n, n= 0..26)
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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 *)
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PROG
| (PARI) a(n)=fibonacci(2*n+1)+fibonacci(2*n-1)
(PARI) a(n)=2*subst(poltchebi(abs(n)), x, 3/2)
(Sage) [lucas_number2(n, 3, 1) for n in range(37)] # Zerinvary Lajos, Jun 25 2008
(PARI) {a(n) = fibonacci(2*n + 1) + fibonacci(2*n - 1)} \\ Michael Somos Aug 11 2009
(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
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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.
A201157
Sequence in context: A131093 A002864 A198634 * A032102 A100388 A186232
Adjacent sequences: A005245 A005246 A005247 * A005249 A005250 A005251
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from James A. Sellers, May 29 2000
Additional comments from Michael Somos, Jun 23 2001
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