Search: id:A025172
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%I A025172
%S A025172 1,1,7,23,17,241,329,1511,5983,1633,57113,99529,314959,1525679,216727,
%T A025172 13297657,28545857,62587199,382087111,200889431,3037005137,7882015153,
%U A025172 11569015927,94076168231,84031193119,678623127841,2113526993753
%V A025172 1,1,-7,-23,17,241,329,-1511,-5983,1633,57113,99529,-314959,-1525679,-216727,
%W A025172 13297657,28545857,-62587199,-382087111,-200889431,3037005137,7882015153,
%X A025172 -11569015927,-94076168231,-84031193119,678623127841,2113526993753
%N A025172 Let phi = arccos(1/3), the dihedral angle of the regular tetrahedron. 
               Then cos(n*phi) = a(n)/3^n.
%C A025172 Used when showing that the regular simplex is not "scisssors-dissectable" 
               to a cube, thus answering Hilbert's third problem.
%D A025172 J. L. Dupont, Scissors Congruences, Group Homology and Characteristic 
               Classes, World Scientific, 2001. See p. 4.
%F A025172 a(0) = 1, a(1) = 1; for n >= 2, a(n) = 2*a(n-1) - 9*a(n-2). - Warut Roonguthai 
               (warut822(AT)yahoo.com), Oct 11 2005
%F A025172 a(n) = (1/2)*(1-2*i*2^(1/2))^(n+1)+(1/2)*(1+2*i*2^(1/2))^(n+1), where 
               i=sqrt(-1). - Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 19 2003
%F A025172 a(n) is the permanent of the matrix M^n, where M = [i, 2; 1, i]. - Simone 
               Severini (simoseve(AT)gmail.com), Apr 27 2007
%p A025172 f:=proc(n) option remember; if n <= 1 then RETURN(1); fi; 2*f(n-1)-9*f(n-2); 
               end;
%t A025172 Table[ n/2 3^n GegenbauerC[ n, 1/3 ], {n, 24} ]
%o A025172 (PARI) {a(n)= if(n<0, 0, 3^(n-1)* subst(3* poltchebi(abs(n)), x, 1/3))} 
               /* Michael Somos Mar 14 2007 */
%Y A025172 Sequence in context: A070411 A167224 A121815 this_sequence A115023 A009228 
               A031450
%Y A025172 Adjacent sequences: A025169 A025170 A025171 this_sequence A025173 A025174 
               A025175
%K A025172 sign
%O A025172 0,3
%A A025172 Wouter Meeussen (wouter.meeussen(AT)pandora.be)
%E A025172 Better description from Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 19 
               2003
%E A025172 Edited by N. J. A. Sloane (njas(AT)research.att.com), Feb 22 2007. Among 
               other things, I changed the offset and the beginning of the sequence, 
               so some of the formulae may need to be adjusted slightly.

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