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%I #22 Sep 08 2022 08:45:47
%S 1,17,227,2743,31441,349241,3802499,40854943,434991553,4602307457,
%T 48477201539,509007338599,5332433173201,55772217368297,
%U 582637691946467,6081473282940943,63438141429166081,661450156372654961
%N a(n) = 18*a(n-1) - 79*a(n-2) for n > 1; a(0) = 1, a(1) = 17.
%C Binomial transform of A081185 without initial term 0. Ninth binomial transform of A164587.
%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 has 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 12 2021
%H G. C. Greubel, <a href="/A164600/b164600.txt">Table of n, a(n) for n = 0..975</a> (terms 0..100 from Vincenzo Librandi)
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (18, -79).
%F a(n) = ((1+4*sqrt(2))*(9+sqrt(2))^n + (1-4*sqrt(2))*(9-sqrt(2))^n)/2.
%F G.f.: (1-x)/(1-18*x+79*x^2).
%F E.g.f.: exp(9*x)*(cosh(sqrt(2)*x) + 4*sqrt(2)*sinh(sqrt(2)*x)). - _G. C. Greubel_, Aug 11 2017
%F From _G. C. Greubel_, Mar 12 2021: (Start)
%F a(n) = 2*A147960(n) + 8*A153593(n).
%F a(n) = (1/2)*Sum_{k=0..n} binomial(n,k)*8^(n-k)*(5*Q(k) + 4*Q(k-1)), where Q(n) = Pell-Lucas(n) = A002203(n). (End)
%p m:=30; S:=series( (1-x)/(1-18*x+79*x^2), x, m+1):
%p seq(coeff(S, x, j), j=0..m); # _G. C. Greubel_, Mar 12 2021
%t LinearRecurrence[{18,-79},{1,17},30] (* _Harvey P. Dale_, Oct 30 2013 *)
%o (Magma) [ n le 2 select 16*n-15 else 18*Self(n-1)-79*Self(n-2): n in [1..18] ];
%o (PARI) my(x='x+O('x^50)); Vec((1-x)/(1-18*x+79*x^2)) \\ _G. C. Greubel_, Aug 11 2017
%o (Sage) [( (1-x)/(1-18*x+79*x^2) ).series(x,n+1).list()[n] for n in (0..30)] # _G. C. Greubel_, Mar 12 2021
%Y Sequences in the class a(n, m): A164298 (m=1), A164299 (m=2), A164300 (m=3), A164301 (m=4), A164598 (m=5), A164599 (m=6), A081185 (m=7), this sequence (m=8).
%Y Cf. A147960, A153593, A164587.
%K nonn
%O 0,2
%A _Klaus Brockhaus_, Aug 17 2009
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