0,1

Drop initial 2; then iterates of A050411 do not diverge for these starting values. - David W. Wilson (davidwwilson(AT)comcast.net)

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

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 (smbelcas(AT)toroidalsnark.net), 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)

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

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.

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]

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]

Tanya Khovanova, Recursive Sequences

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, 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

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 (se16(AT)btinternet.com), May 08 2001

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

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 (dwalsh(AT)mtsu.edu), 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 (pbarry(AT)wit.ie), Apr 24 2004

a(n)={floor((3+sqrt(5))^n) + 1}/2^n. - Lekraj Beedassy (blekraj(AT)yahoo.com), Oct 22 2004

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

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

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]

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]

Contribution from Sarah-Marie Belcastro (smbelcas(AT)toroidalsnark.net), Jul 04 2009: (Start)

(L_{2n}-F_{2n})/2+F_{2n+1} (Tesler)

\prod_{r=1}^{n}(1+4\sin^2(((4r-1)\pi)/(4n))) (Lu/Wu) (End)

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

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 (heinz(AT)hs-heilbronn.de), Jul 31 2008]

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 (smbelcas(AT)toroidalsnark.net), Jul 04 2009]

(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 (zerinvarylajos(AT)yahoo.com), Jun 25 2008

(PARI) {a(n) = fibonacci(2*n + 1) + fibonacci(2*n - 1)} - Michael Somos Aug 11 2009

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.

Sequence in context: A058334 A131093 A002864 this_sequence A032102 A100388 A160181

Adjacent sequences: A005245 A005246 A005247 this_sequence A005249 A005250 A005251

nonn,easy

N. J. A. Sloane (njas(AT)research.att.com).

More terms from James A. Sellers (sellersj(AT)math.psu.edu), May 29 2000

Additional comments from Michael Somos, Jun 23 2001