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%I #39 Jan 01 2024 11:44:55
%S 1,9,65,441,2929,19305,126881,833049,5467345,35877321,235418369,
%T 1544728185,10135859761,66507086889,436390025825,2863396842201,
%U 18788331166609,123280631024265,808912380552641,5307721328585529
%N a(n) = 9*a(n-1) - 16*a(n-2), with a(0) = 1, a(1) = 9.
%H Indranil Ghosh, <a href="/A102902/b102902.txt">Table of n, a(n) for n = 0..1221</a>
%H R. Flórez, R. A. Higuita, and A. Mukherjee, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Mukherjee/mukh2.html">Alternating Sums in the Hosoya Polynomial Triangle</a>, Article 14.9.5, Journal of Integer Sequences, Vol. 17 (2014).
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (9,-16).
%F G.f.: 1/(1-9*x+16*x^2).
%F a(n) = Sum_{k=0..n} binomial(2*n-k+1, k)*4^k.
%F a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-16)^k*9^(n-2*k).
%F a(n) = 4^n * ChebyshevU(n, 9/8). - _G. C. Greubel_, Dec 09 2022
%t LinearRecurrence[{9,-16},{1,9},20] (* _Harvey P. Dale_, Jul 28 2016 *)
%o (SageMath) [lucas_number1(n,9,16) for n in range(1, 21)] # _Zerinvary Lajos_, Apr 23 2009
%o (Magma) [4^n*Evaluate(ChebyshevSecond(n+1), 9/8): n in [0..30]]; // _G. C. Greubel_, Dec 09 2022
%Y Cf. A002540, A099459.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Jan 17 2005
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