This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A025170 #48 Jan 02 2024 22:48:23
%S A025170 1,-2,-5,28,-11,-230,559,952,-6935,5302,51811,-151340,-163619,1689298,
%T A025170 -1906025,-11391632,39937489,22649710,-404736821,605626252,2431378885,
%U A025170 -10313394038,-1255621889,95331790120,-179362983239,-499260144602,2612787138355,-732232975292
%N A025170 Expansion of g.f.: 1/(1 + 2*x + 9*x^2).
%C A025170 Reciprocal Chebyshev polynomial of second kind evaluated at 3 multiplied by (-1)^n.
%C A025170 From Sharon Sela (sharonsela(AT)hotmail.com), Jan 19 2002: (Start)
%C A025170 a(n) is (-1)^n times the determinant of the following tridiagonal n X n matrix:
%C A025170 [2 3 0 . . . . . . .]
%C A025170 [3 2 3 0 . . . . . .]
%C A025170 [0 3 2 3 0 . . . . .]
%C A025170 [. 0 3 2 3 0 . . . .]
%C A025170 [. . . . . . . . . .]
%C A025170 [. . . . . . . . . .]
%C A025170 [. . . . 0 3 2 3 0 .]
%C A025170 [. . . . . 0 3 2 3 0]
%C A025170 [. . . . . . 0 3 2 3]
%C A025170 [. . . . . . . 0 3 2]
%C A025170 (End)
%H A025170 G. C. Greubel, <a href="/A025170/b025170.txt">Table of n, a(n) for n = 0..1000</a>
%H A025170 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-2,-9).
%H A025170 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A025170 a(n) = 3^n * ChebyshevU(n, -1/3).
%F A025170 a(n) = ( A088137(n+1) )^2 + ( A087455(n+1)/2 )^2 - ( A087455(n+2)/2 )^2. - _Creighton Dement_, Aug 20 2004
%F A025170 a(n) = -(2*a(n-1) + 9*a(n-2)) for n>1, with a(0)=1, a(1)=-2. - _Philippe Deléham_, Sep 19 2009
%F A025170 a(n) = (-2)^n*Product_{k=1..n}(1 + 3*cos(k*Pi/(n+1))). - _Peter Luschny_, Nov 28 2019
%F A025170 From _G. C. Greubel_, Jan 02 2024: (Start)
%F A025170 a(n) = (-1)^n * A127357(n).
%F A025170 E.g.f.: (1/4)*exp(-x)*(4*cos(2*sqrt(2)*x) - sqrt(2)*sin(2*sqrt(2)*x)). (End)
%t A025170 Table[3^n ChebyshevU[n, -1/3], {n, 0, 24}]
%o A025170 (PARI) a(n)=if(n<0,0,polcoeff(1/(1+2*x+9*x^2)+x*O(x^n),n))
%o A025170 (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 */
%o A025170 (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 */
%o A025170 (Magma) [(-3)^n*Evaluate(ChebyshevU(n+1),1/3): n in [0..50]]; // _G. C. Greubel_, Jan 02 2024
%o A025170 (SageMath) [3^n*chebyshev_U(n,-1/3) for n in range(41)] # _G. C. Greubel_, Jan 02 2024
%Y A025170 Variant is A127357.
%Y A025170 Cf. A087455, A088137.
%K A025170 sign,easy
%O A025170 0,2
%A A025170 _Wouter Meeussen_