cp's OEIS Frontend

A102902 a(n) = 9*a(n-1) - 16*a(n-2), with a(0) = 1, a(1) = 9.

%I A102902 #45 Jul 27 2025 11:15:51
%S A102902 1,9,65,441,2929,19305,126881,833049,5467345,35877321,235418369,
%T A102902 1544728185,10135859761,66507086889,436390025825,2863396842201,
%U A102902 18788331166609,123280631024265,808912380552641,5307721328585529
%N A102902 a(n) = 9*a(n-1) - 16*a(n-2), with a(0) = 1, a(1) = 9.
%H A102902 Indranil Ghosh, <a href="/A102902/b102902.txt">Table of n, a(n) for n = 0..1221</a>
%H A102902 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 A102902 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (9,-16).
%F A102902 G.f.: 1/(1-9*x+16*x^2).
%F A102902 a(n) = Sum_{k=0..n} binomial(2*n-k+1, k)*4^k.
%F A102902 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-16)^k*9^(n-2*k).
%F A102902 a(n) = 4^n * ChebyshevU(n, 9/8). - _G. C. Greubel_, Dec 09 2022
%F A102902 From _Peter Bala_, Jul 23 2025: (Start)
%F A102902 a(n) := ((9 + sqrt(17))^(n+1) - (9 - sqrt(17))^(n+1))/(2^(n+1)*sqrt(17)).
%F A102902 The following products telescope:
%F A102902 Product_{k >= 1} 1 + 4^k/a(k) = (1 + sqrt(17))/2.
%F A102902 Product_{k >= 1} 1 - 4^k/a(k) = (1 + sqrt(17))/18.
%F A102902 Product_{k >= 1} 1 + (-4)^k/a(k) = (17 + sqrt(17))/34.
%F A102902 Product_{k >= 1} 1 - (-4)^k/a(k) = (17 + sqrt(17))/18. (End)
%t A102902 LinearRecurrence[{9,-16},{1,9},20] (* _Harvey P. Dale_, Jul 28 2016 *)
%o A102902 (SageMath) [lucas_number1(n,9,16) for n in range(1, 21)] # _Zerinvary Lajos_, Apr 23 2009
%o A102902 (Magma) [4^n*Evaluate(ChebyshevSecond(n+1), 9/8): n in [0..30]]; // _G. C. Greubel_, Dec 09 2022
%Y A102902 Cf. A002450, A004187, A016153, A099459.
%K A102902 easy,nonn
%O A102902 0,2
%A A102902 _Paul Barry_, Jan 17 2005