0,2

Solves for y in x^2 - 3*y^2 = 4. Quadruples (a=b-y, b, c=b+y, d), with b=y^2 + 1 and d=x*y, where (x, y) solves x^2 - 3*y^2 = 4, verify the triangle relation (a^2 + b^2 + c^2 + d^2)^2 = 3*(a^4 + b^4 + c^4 + d^4). Thus d corresonds to the distance sum of the Fermat(or Torriccelli)point from its vertices in a triangle whose sides are in A.P. with middle side b, and common difference y.

Tanya Khovanova, Recursive Sequences

a(n)=-(1/3)*sqrt(3)*[7-4*sqrt(3)]^n+(1/3)*sqrt(3)*[7+4*sqrt(3)]^n , with n>=0 [From Paolo P. Lava (ppl(AT)spl.at), Oct 02 2008]

a := proc(n) option remember: if n=0 then RETURN(0) fi: if n=1 then RETURN(8) fi: 14*a(n-1)-a(n-2): end: for n from 0 to 30 do printf(`%d, `, a(n)) od:

Cf. A067902.

First differences of A045899.

Equals 8 * A007655(n+1).

Adjacent sequences: A067897 A067898 A067899 this_sequence A067901 A067902 A067903

Sequence in context: A075851 A053536 A139741 this_sequence A067414 A034689 A010041

nonn,new

Lekraj Beedassy (blekraj(AT)yahoo.com), May 13 2003

More terms from Ray Chandler (rayjchandler(AT)sbcglobal.net) and James A. Sellers (sellersj(AT)math.psu.edu), May 19, 2003