A067900 a(n) = 14*a(n-1) - a(n-2); a(0) = 0, a(1) = 8.

0, 8, 112, 1560, 21728, 302632, 4215120, 58709048, 817711552, 11389252680, 158631825968, 2209456310872, 30773756526240, 428623135056488, 5969950134264592, 83150678744647800, 1158139552290804608, 16130803053326616712, 224673103194281829360, 3129292641666618994328

Offset: 0

Author

Lekraj Beedassy, May 13 2003

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A067902.

First differences of A045899.

Equals 8 * A007655(n+1).

Programs

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

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

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 *)

PARI

vector(21, n, 8*polchebyshev(n-2, 2, 7) ) \\ G. C. Greubel, Dec 23 2019

Sage

[8*chebyshev_U(n-1,7) for n in (0..20)] # G. C. Greubel, Dec 23 2019

Formula

G.f.: 8*x/(1-14*x+x^2). - Philippe Deléham, Nov 17 2008

E.g.f.: 2*exp(7*x)*sinh(4*sqrt(3)*x)/sqrt(3). - Stefano Spezia, Dec 12 2022