0,1

a(n) gives values of x satisfying x^2 - 3*y^2 = 4; corresponding y values are given by 2*A001353(n).

If M is any given term of the sequence, then the next one is 2M + sqrt(3M^2 - 12). - Lekraj Beedassy (blekraj(AT)yahoo.com), Feb 18 2002

For n>0, a(n)-1, a(n), a(n)+1 form a Fleenor-Heronian triangle, i.e. a Heronian triangle with consecutive sides, whose area A(n) may be obtained from the relation [4*A(n)]^2 = 3([a(2n)]^2 - 4); or A(n)=3*A001353(2n)/2, and whose semipermeter = 3*a[n]/2. The sequence is symmetrical about a[0], i.e.; a[ -n] = a[n].

For n>0, a(n)+2 is the number of dimer tilings of a 2n x 2 Klein bottle (cf. A103999).

The terms whose index is a power of 2 form A003010. - John Blythe Dobson (j.dobson(AT)uwinnipeg.ca), Oct 28 2007

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.

R. A. Beauregard and E. R. Suryanarayan, The Brahmagupta Triangles, The College Mathematics Journal 29(1) 13-7 1998 MAA.

B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 82.

Michael P. Cohen, Generating Heronian Triangles With Consecutive Integer Sides. Journal of Recreational Mathematics, vol. 30 no. 2 1999-2000 p. 123.

L. E. Dickson, History of The Theory of Numbers, Vol. 2 pp. 197;198;200;201. Chelsea NY.

Charles R. Fleenor, Heronian Triangles with Consecutive Integer Sides, Journal of Recreational Mathematics, Volume 28, no. 2 (1996-7) 113-115.

H. W. Gould, A triangle with integral sides and area, Fib. Quart., 11 (1973), 27-39.

E. K. Lloyd, "The standard deviation of 1, 2, .., n, Pell's equation and rational triangles", preprint.

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

V. D. To, "Finding All Fleenor-Heronian Triangles", Journal of Recreational Mathematics vol. 32 no.4 2003-4 pp. 298-301 Baywood NY.

T. D. Noe, Table of n, a(n) for n=0..200

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.

K. S. Brown's Mathpages, Some Properties of the Lucas Sequence(2, 4, 14, 52, 194, ...)

Tanya Khovanova, Recursive Sequences

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

a(n) = ( 2 + Sqrt(3) )^n + ( 2 - Sqrt(3) )^n.

a(n) = trace of (n+1)st power of the 2 X 2 matrix [1 2 / 1 3]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 30 2003

From the addition formula, a(n+m) = a(n)*a(m) - a(m-n), it is easy to derive multiplication formulae, such as: a(2n) = (a(n))^2 - 2, a(3n) = (a(n))^3 - 3*(a(n)), a(4n) = (a(n))^4 - 4*(a(n))^2 + 2, a(5n) = (a(n))^5 - 5*(a(n))^3 + 5*(a(n)), a(6n) = (a(n))^6 - 6*(a(n))^4 + 9*(a(n))^2 - 2, etc. The absolute values of the coefficients in the expansions are given by the triangle A034807. - John Blythe Dobson (j.dobson(AT)uwinnipeg.ca), Nov 04 2007

G.f.: -2*(-1+2*x)/(1-4*x+x^2). a(n)=2*A001353(n+1)-4*A001353(n). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 16 2007

A003500 := proc(n) option remember; if n <= 1 then 2*n+2 else 4*A003500(n-1)-A003500(n-2); fi; end;

A003500:=-2*(-1+2*z)/(1-4*z+z**2); [Conjectured by S. Plouffe in his 1992 dissertation.]

a[0] = 2; a[1] = 4; a[n_] := a[n] = 4a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 23}]

Equals A001353(n+1) - A001353(n-1), also A001835(n) + A001835(n+1), also 2*A001075(n).

Cf. A001570, A006051, A048788, A002530, A011945.

Adjacent sequences: A003497 A003498 A003499 this_sequence A003501 A003502 A003503

Sequence in context: A032222 A046650 A055727 this_sequence A129876 A038055 A006385

nonn,easy,nice

njas

More terms from James A. Sellers (sellersj(AT)math.psu.edu), May 03 2000

Additional comments from Lekraj Beedassy (blekraj(AT)yahoo.com), Feb 14 2002