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Let phi = arccos(1/3), the dihedral angle of the regular tetrahedron. Then cos(n*phi) = a(n)/3^n.
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%I #57 Apr 07 2022 06:42:41

%S 1,1,-7,-23,17,241,329,-1511,-5983,1633,57113,99529,-314959,-1525679,

%T -216727,13297657,28545857,-62587199,-382087111,-200889431,3037005137,

%U 7882015153,-11569015927,-94076168231,-84031193119,678623127841,2113526993753

%N Let phi = arccos(1/3), the dihedral angle of the regular tetrahedron. Then cos(n*phi) = a(n)/3^n.

%C Used when showing that the regular simplex is not "scisssors-dissectable" to a cube, thus answering Hilbert's third problem.

%C From _Peter Bala_, Apr 01 2018: (Start)

%C This sequence is (1/2) * the Lucas sequence V(n,2,9). The companion Lucas sequence U(n,2,9) is A127357.

%C Define a binary operation o on rational numbers by x o y = (x + y)/(1 - 2*x*y). This is a commutative and associative operation with identity 0. Then 2 o 2 o ... o 2 (n terms) = 2*A127357(n-1)/A025172(n). Cf. A088137 and A087455. (End)

%H Vincenzo Librandi, <a href="/A025172/b025172.txt">Table of n, a(n) for n = 0..1000</a>

%H J. L. Dupont, <a href="http://www.worldscientific.com/worldscibooks/10.1142/4598">Scissors Congruences, Group Homology and Characteristic Classes</a>, World Scientific, 2001. See p. 4.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lucas_sequence">Lucas sequence</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-9).

%F a(0) = 1, a(1) = 1; for n >= 2, a(n) = 2*a(n-1) - 9*a(n-2). - _Warut Roonguthai_, Oct 11 2005

%F a(n) = (1/2)*(1-2*i*2^(1/2))^n+(1/2)*(1+2*i*2^(1/2))^n, where i=sqrt(-1). - _Vladeta Jovovic_, Apr 19 2003

%F a(n) is the permanent of the matrix M^n, where M = [i, 2; 1, i]. - _Simone Severini_, Apr 27 2007

%F a(n) = Product_{i=1..n} (2 - tan((i-1/2)*Pi/(2*n))^2). - _Gerry Martens_, May 26 2011

%F G.f.: (1-x)/(1-2*x+9*x^2). - _Colin Barker_, Jun 21 2012

%F G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(8*k+1)/(x*(8*k+9) + 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 28 2013

%F E.g.f.: exp(x)*cos(2*sqrt(2)*x). - _Vladimir Reshetnikov_, Oct 15 2016

%F a(n) = A127357(n)-A127357(n-1). - _R. J. Mathar_, Apr 07 2022

%p f:=proc(n) option remember; if n <= 1 then RETURN(1); fi; 2*f(n-1)-9*f(n-2); end;

%t Table[ n/2 3^n GegenbauerC[ n, 1/3 ], {n, 24} ]

%t CoefficientList[Series[(1 - x)/(1 - 2 x + 9 x^2), {x, 0, 30}], x] (* _Vincenzo Librandi_, Oct 17 2013 *)

%t LinearRecurrence[{2,-9},{1,1},30] (* _Harvey P. Dale_, Jan 30 2016 *)

%o (PARI) {a(n)= if(n<0, 0, 3^(n-1)* subst(3* poltchebi(abs(n)), x, 1/3))} /* _Michael Somos_, Mar 14 2007 */

%Y Cf. A088137, A087455, A127357.

%K sign,easy

%O 0,3

%A _Wouter Meeussen_

%E Better description from _Vladeta Jovovic_, Apr 19 2003

%E Edited by _N. J. A. Sloane_, Feb 22 2007. Among other things, I changed the offset and the beginning of the sequence, so some of the formulas may need to be adjusted slightly.