A025172 Let phi = arccos(1/3), the dihedral angle of the regular tetrahedron. Then cos(n*phi) = a(n)/3^n.

1, 1, -7, -23, 17, 241, 329, -1511, -5983, 1633, 57113, 99529, -314959, -1525679, -216727, 13297657, 28545857, -62587199, -382087111, -200889431, 3037005137, 7882015153, -11569015927, -94076168231, -84031193119, 678623127841, 2113526993753

Offset

0,3

Comments

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

From Peter Bala, Apr 01 2018: (Start)

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

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)

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

J. L. Dupont, Scissors Congruences, Group Homology and Characteristic Classes, World Scientific, 2001. See p. 4.

Wikipedia, Lucas sequence

Index entries for linear recurrences with constant coefficients, signature (2,-9).

Formula

a(0) = 1, a(1) = 1; for n >= 2, a(n) = 2*a(n-1) - 9*a(n-2). - Warut Roonguthai, Oct 11 2005

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

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

a(n) = Product_{i=1..n} (2 - tan((i-1/2)*Pi/(2*n))^2). - Gerry Martens, May 26 2011

G.f.: (1-x)/(1-2*x+9*x^2). - Colin Barker, Jun 21 2012

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

E.g.f.: exp(x)*cos(2*sqrt(2)*x). - Vladimir Reshetnikov, Oct 15 2016

a(n) = A127357(n)-A127357(n-1). - R. J. Mathar, Apr 07 2022

Maple

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

Mathematica

Table[ n/2 3^n GegenbauerC[ n, 1/3 ], {n, 24} ]
CoefficientList[Series[(1 - x)/(1 - 2 x + 9 x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 17 2013 *)
LinearRecurrence[{2, -9}, {1, 1}, 30] (* Harvey P. Dale, Jan 30 2016 *)

Prog

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

Crossrefs

Cf. A088137, A087455, A127357.

Sequence in context: A167224 A175483 A121815 * A115023 A009228 A249438

Adjacent sequences: A025169 A025170 A025171 * A025173 A025174 A025175

Keyword

sign,easy

Author

Wouter Meeussen

Extensions

Better description from Vladeta Jovovic, Apr 19 2003

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.

Status

approved