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%I #44 Dec 16 2023 17:44:45
%S 0,8,112,1560,21728,302632,4215120,58709048,817711552,11389252680,
%T 158631825968,2209456310872,30773756526240,428623135056488,
%U 5969950134264592,83150678744647800,1158139552290804608,16130803053326616712,224673103194281829360,3129292641666618994328
%N a(n) = 14*a(n-1) - a(n-2); a(0) = 0, a(1) = 8.
%C 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.
%H G. C. Greubel, <a href="/A067900/b067900.txt">Table of n, a(n) for n = 0..870</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (14,-1).
%F G.f.: 8*x/(1-14*x+x^2). - _Philippe Deléham_, Nov 17 2008
%F E.g.f.: 2*exp(7*x)*sinh(4*sqrt(3)*x)/sqrt(3). - _Stefano Spezia_, Dec 12 2022
%p 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:
%p seq( simplify(8*ChebyshevU(n-1, 7)), n=0..20); # _G. C. Greubel_, Dec 23 2019
%t LinearRecurrence[{14,-1}, {0,8}, 17] (* _Jean-François Alcover_, Sep 19 2017 *)
%t 8*ChebyshevU[Range[21] -2, 7] (* _G. C. Greubel_, Dec 23 2019 *)
%o (PARI) vector(21, n, 8*polchebyshev(n-2, 2, 7) ) \\ _G. C. Greubel_, Dec 23 2019
%o (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
%o (Sage) [8*chebyshev_U(n-1,7) for n in (0..20)] # _G. C. Greubel_, Dec 23 2019
%o (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
%Y Cf. A067902.
%Y First differences of A045899.
%Y Equals 8 * A007655(n+1).
%K nonn,easy
%O 0,2
%A _Lekraj Beedassy_, May 13 2003
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