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 A025170 G.f.: 1/(1+2x+9x^2). 2
 1, -2, -5, 28, -11, -230, 559, 952, -6935, 5302, 51811, -151340, -163619, 1689298, -1906025, -11391632, 39937489, 22649710, -404736821, 605626252, 2431378885, -10313394038, -1255621889, 95331790120, -179362983239, -499260144602, 2612787138355, -732232975292 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Reciprocal Chebyshev polynomial of second kind evaluated at 3 multiplied by (-1)^n. a(n) is (-1)^n times the determinant of the following tri-diagonal n X n matrix : [2 3 0 0 ... ] [3 2 3 0 ... ] [0 3 2 3 0 ... ] [. 0 3 2 3 ... ] [. . . . . ] [. . . 3 2 3 0] [. . . 0 3 2 3] [. . . 0 0 3 2] - Sharon Sela (sharonsela(AT)hotmail.com), Jan 19 2002 LINKS Index entries for linear recurrences with constant coefficients, signature (-2,-9). FORMULA a(n) = ( A088137(n+1) )^2 + ( A087455(n+1)/2 )^2 - ( A087455(n+2)/2 )^2. - Creighton Dement, Aug 20 2004 A025170(n) = ( A088137(n+1) )^2 + ( A087455(n+1)/2 )^2 - ( A087455(n+2)/2 )^2. Using the known formula ( see A088137 ) |3*A087455(n) - A087455(n+1)| = 2*A088137(n+1) or 3*A087455(n) + A087455(n+1) = 2*A088137(n+1) A025170 can be expressed entirely using A087455 - Creighton Dement, Aug 22 2004 a(0)=1, a(1)=-2, a(n) = -(2*a(n-1)+9*a(n-2)) for n>1. [From Philippe Deléham, Sep 19 2009] MATHEMATICA Table[ 3^n ChebyshevU[ n, -1/3 ], {n, 0, 24} ] PROG (PARI) a(n)=if(n<0, 0, polcoeff(1/(1+2*x+9*x^2)+x*O(x^n), n)) (PARI) a(n)=if(n<0, 0, 3^n*subst(poltchebi(n+1)+3*poltchebi(n), 'x, -1/3)*3/8) /* Michael Somos, Sep 15 2005 */ (PARI) a(n)=if(n<0, 0, (-1)^n*matdet(matrix(n, n, i, j, if(abs(i-j)<2, 2+abs(i-j))))) /* Michael Somos, Sep 15 2005 */ CROSSREFS Cf. A087455, A088137. Sequence in context: A208224 A208227 A127357 * A151775 A286879 A095159 Adjacent sequences:  A025167 A025168 A025169 * A025171 A025172 A025173 KEYWORD sign,easy AUTHOR STATUS approved

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