%I #24 Sep 08 2022 08:45:47

%S 1,14,134,1132,9028,69848,531224,3999856,29936656,223244768,

%T 1661090912,12342768832,91636134976,679979479424,5044125163904,

%U 37410199666432,277422140424448,2057118896434688,15253073982785024,113094845314640896

%N a(n) = 12*a(n-1) - 34*a(n-2), for n > 1, with a(0) = 1, a(1) = 14.

%C Binomial transform of A164301. Sixth binomial transform of A164587. Inverse binomial transform of A164599.

%C This sequence is part of a class of sequences defined by the recurrence a(n,m) = 2*(m+1)*a(n-1,m) - ((m+1)^2 -2)*a(n-2,m) with a(0) = 1 and a(1) = m+9. The generating function is Sum_{n>=0} a(n,m)*x^n = (1 - (m-7)*x)/(1 - 2*(m+1)*x + ((m+1)^2 -2)*x^2) and have a series expansion in terms of Pell-Lucas numbers defined by a(n, m) = (1/2)*Sum_{k=0..n} binomial(n,k)*m^(n-k)*(5*Q(k) + 4*Q(k-1)). - _G. C. Greubel_, Mar 11 2021

%H G. C. Greubel, <a href="/A164598/b164598.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..100 from Vincenzo Librandi)

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (12,-34).

%F a(n) = ((1+4*sqrt(2))*(6+sqrt(2))^n + (1-4*sqrt(2))*(6-sqrt(2))^n)/2.

%F G.f.: (1+2*x)/(1-12*x+34*x^2).

%F E.g.f.: exp(6*x)*(cosh(sqrt(2)*x) + 4*sqrt(2)*sinh(sqrt(2)*x)). - _G. C. Greubel_, Aug 11 2017

%F From _G. C. Greubel_, Mar 11 2021: (Start)

%F a(n) = A147957(n) + 8*A081183(n).

%F a(n) = (1/2)*Sum_{k=0..n} binomial(n,k)*5^(n-k)*(5*Q(k) + 4*Q(k-1)), where Q(n) = Pell-Lucas(n) = A002203(n). (End)

%p m:=30; S:=series( (1+2*x)/(1-12*x+34*x^2), x, m+1):

%p seq(coeff(S, x, j), j=0..m); # _G. C. Greubel_, Mar 11 2021

%t LinearRecurrence[{12,-34}, {1,14}, 30] (* _G. C. Greubel_, Aug 11 2017 *)

%o (Magma) [ n le 2 select 13*n-12 else 12*Self(n-1)-34*Self(n-2): n in [1..30] ];

%o (PARI) my(x='x+O('x^30)); Vec((1+2*x)/(1-12*x+34*x^2)) \\ _G. C. Greubel_, Aug 11 2017

%o (Sage) [( (1+2*x)/(1-12*x+34*x^2) ).series(x,n+1).list()[n] for n in (0..30)] # _G. C. Greubel_, Mar 11 2021

%Y Sequences in the class a(n, m): A164298 (m=1), A164299 (m=2), A164300 (m=3), A164301 (m=4), this sequence (m=5), A164599 (m=6), A081185 (m=7), A164600 (m=8).

%Y Cf. A081183, A147957, A164587.

%K nonn,easy

%O 0,2

%A _Klaus Brockhaus_, Aug 17 2009