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 A067900 a(n) = 14*a(n-1) - a(n-2); a(0) = 0, a(1) = 8. 6
 0, 8, 112, 1560, 21728, 302632, 4215120, 58709048, 817711552, 11389252680, 158631825968, 2209456310872, 30773756526240, 428623135056488, 5969950134264592, 83150678744647800, 1158139552290804608 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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, satisfy the triangle relation (a^2 + b^2 + c^2 + d^2)^2 = 3*(a^4 + b^4 + c^4 + d^4). Thus d corresponds 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. LINKS G. C. Greubel, Table of n, a(n) for n = 0..870 Tanya Khovanova, Recursive Sequences Index entries for linear recurrences with constant coefficients, signature (14,-1). FORMULA a(n) = ( (7+4*sqrt(3))^n - (7-4*sqrt(3))^n )/sqrt(3), with n>=0. - Paolo P. Lava, Oct 02 2008 G.f.: 8*x/(1-14*x+x^2). - Philippe Deléham, Nov 17 2008 MAPLE 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: seq( simplify(8*ChebyshevU(n-1, 7)), n=0..20); # G. C. Greubel, Dec 23 2019 MATHEMATICA LinearRecurrence[{14, -1}, {0, 8}, 17] (* Jean-François Alcover, Sep 19 2017 *) 8*ChebyshevU[Range[21] -2, 7] (* G. C. Greubel, Dec 23 2019 *) PROG (PARI) vector(21, n, 8*polchebyshev(n-2, 2, 7) ) \\ G. C. Greubel, Dec 23 2019 (MAGMA) m:=7; I:=[0, 8]; [n le 2 select I[n] else 2*m*Self(n-1) -Self(n-2): n in [1..20]]; // G. C. Greubel, Dec 23 2019 (Sage) [8*chebyshev_U(n-1, 7) for n in (0..20)] # G. C. Greubel, Dec 23 2019 (GAP) m:=7;; a:=[0, 8];; for n in [3..20] do a[n]:=2*m*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Dec 23 2019 CROSSREFS Cf. A067902. First differences of A045899. Equals 8 * A007655(n+1). Sequence in context: A270111 A053536 A139741 * A067414 A265665 A302104 Adjacent sequences:  A067897 A067898 A067899 * A067901 A067902 A067903 KEYWORD nonn,easy AUTHOR Lekraj Beedassy, May 13 2003 STATUS approved

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Last modified April 13 00:24 EDT 2021. Contains 342934 sequences. (Running on oeis4.)