0,1

Two times Chebyshev polynomials of the first kind evaluated at 3.

Also 2(a(2n)-2) and a(2n+1)-2 are perfect squares. - Mario Catalani (mario.catalani(AT)unito.it), Mar 31 2003

Chebyshev polynomials of the first kind evaluated at 3, then multiplied by 2. - Michael Somos, Apr 07 2003

Also gives solutions >2 to the equation x^2-3 = floor(x*r*floor(x/r)) where r=sqrt(2) - Benoit Cloitre (benoit7848c(AT)orange.fr), Feb 14 2004

Output of Lu and Wu's formula for the number of perfect matchings of an m x n Klein bottle where m and n are both even specializes to this sequence for m=2. [From Sarah-Marie Belcastro (smbelcas(AT)toroidalsnark.net), Jul 04 2009]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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.

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 198.

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

Thomas Koshy, Fibonacci and Lucas Numbers with Applications, 2001, Wiley, p. 77-79.

Jay Kappraff, Beyond Measure, A Guided Tour Through Nature, Myth and Number, World Scientific, 2002; p. 480-481.

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]

Index entries for two-way infinite sequences

Index entries for sequences related to linear recurrences with constant coefficients

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.

Tanya Khovanova, Recursive Sequences

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

G.f.: (2-6x)/(1-6x+x^2). a(n)=(3+2*sqrt(2))^n+(3-2*sqrt(2))^n.

For all sequence elements n, 2*n^2 - 8 is a perfect square. Lim a(n)/a(n-1) = 3 + 2*sqrt(2) - Gregory V. Richardson (omomom(AT)hotmail.com), Oct 06 2002

a(2n)+2 is a perfect square, 2(a(2n+1)+2) is a perfect square. a(n), a(n-1) and A077445(n), n>0, satisfy the Diophantine equation x^2+y^2-3z^2=-8. - Mario Catalani (mario.catalani(AT)unito.it), Mar 24 2003

a(n+1)=Trace of n-th power of matrix {{6, -1}, {1, 0}} - Artur Jasinski (grafix(AT)csl.pl), Apr 22 2008

\prod_{r=1}^{n}(4\sin^2((4r-1)\pi/(4n))+4) (Lu/Wu) [From Sarah-Marie Belcastro (smbelcas(AT)toroidalsnark.net), Jul 04 2009]

A003499:=-2*(-1+3*z)/(1-6*z+z**2); [Conjectured by S. Plouffe in his 1992 dissertation.]

a[0] = 2; a[1] = 6; a[n_] := 6a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 19}] (from Robert G. Wilson v Jan 30 2004)

Table[Tr[MatrixPower[{{6, -1}, {1, 0}}, n]], {n, 1, 100}] - Artur Jasinski (grafix(AT)csl.pl), Apr 22 2008

(PARI) a(n)=2*real((3+quadgen(32))^n)

(PARI) a(n)=2*subst(poltchebi(abs(n)), x, 3)

(PARI) a(n)=if(n<0, a(-n), polsym(1-6*x+x^2, n)[n+1])

sage: [lucas_number2(n, 6, 1) for n in range(37)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2008

a(n)=2 A001541(n). A081555(n)=1+a(n).

Bisection of A002203.

First row of array A103999.

Adjacent sequences: A003496 A003497 A003498 this_sequence A003500 A003501 A003502

Sequence in context: A026976 A026951 A030233 this_sequence A018953 A009199 A052824

nonn

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