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A157335 Expansion of 1/( (1+x)*(1-7*x+x^2) ).

%I A157335 #40 Jun 12 2024 06:56:23
%S A157335 1,6,42,287,1968,13488,92449,633654,4343130,29768255,204034656,
%T A157335 1398474336,9585285697,65698525542,450304393098,3086432226143,
%U A157335 21154721189904,144996616103184,993821591532385,6811754524623510,46688460080832186,320007466041201791
%N A157335 Expansion of 1/( (1+x)*(1-7*x+x^2) ).
%H A157335 Colin Barker, <a href="/A157335/b157335.txt">Table of n, a(n) for n = 0..1000</a>
%H A157335 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Barry4/barry64.html">Symmetric Third-Order Recurring Sequences, Chebyshev Polynomials, and Riordan Arrays</a>, JIS 12 (2009) 09.8.6
%H A157335 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,6,-1).
%F A157335 a(n) = A152119(n+1) * A152119(n+2).
%F A157335 a(n) = Sum_{k=0..floor(n/2)} (-1)^(n-k)*U(k,7/2) where U(n,x) is the n-th Chebyshev polynomial of the second kind;
%F A157335 a(n) = Sum_{k=0..2n} F(n+1-floor(k/2))*F(n+1-(k mod 2)-floor(k/2))*A152119(k+1).
%F A157335 a(n) = 7*a(n-1) - a(n-2) + (-1)^n, n > 1. - _Vincenzo Librandi_, Mar 13 2011
%F A157335 9*a(n) = 8*A004187(n+1) - A004187(n) + (-1)^n. - _R. J. Mathar_, Mar 15 2011
%F A157335 a(n)= ((-1)^(n+1)/3)*Sum_{k=0..n+1} (-1)^k*Fibonacci(4*k). - _Gary Detlefs_, Jan 21 2013
%F A157335 a(n) = 6*a(n-1) + 6*a(n-2) - a(n-3), with a(0)=1, a(1)=6, a(2)=42. - _Harvey P. Dale_, Oct 25 2014
%F A157335 a(n) = (2^(-n)*(5*(-2)^n + (20-9*sqrt(5))*(7-3*sqrt(5))^n + (7+3*sqrt(5))^n*(20+9*sqrt(5))))/45. - _Colin Barker_, Mar 05 2016
%t A157335 CoefficientList[Series[1/((1+x)(1-7x+x^2)),{x,0,20}],x] (* or *) LinearRecurrence[{6,6,-1},{1,6,42},20] (* _Harvey P. Dale_, Oct 25 2014 *)
%o A157335 (PARI) Vec(1/(x^3-6*x^2-6*x+1) + O(x^30)) \\ _Colin Barker_, Mar 05 2016
%o A157335 (Magma) I:=[0, 6, 42]; [n le 3 select I[n] else 6*Self(n-1) + 6*Self(n-2) - Self(n-3): n in [1..30]]; // _G. C. Greubel_, Feb 01 2018
%Y A157335 Cf. A004187, A152119.
%K A157335 easy,nonn
%O A157335 0,2
%A A157335 _Paul Barry_, Feb 27 2009