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%I #20 Sep 08 2022 08:45:47
%S 1,13,107,771,5249,34757,226843,1469019,9472801,60940573,391531307,
%T 2513679891,16131578849,103501150997,663985196443,4259325491499,
%U 27321595396801,175251467663533,1124117982508907,7210396068827811,46249247090573249,296653361322692837
%N a(n) = ((1+4*sqrt(2))*(5+sqrt(2))^n + (1-4*sqrt(2))*(5-sqrt(2))^n)/2.
%C Binomial transform of A164300. Fifth binomial transform of A164587. Inverse binomial transform of A164598.
%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="/A164301/b164301.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 (10,-23).
%F a(n) = 10*a(n-1) - 23*a(n-2) for n > 1; a(0) = 1, a(1) = 13.
%F G.f.: (1+3*x)/(1-10*x+23*x^2).
%F E.g.f.: ( cosh(sqrt(2)*x) + 4*sqrt(2)*sinh(sqrt(2)*x) )*exp(5*x). - _G. C. Greubel_, Sep 13 2017
%F From _G. C. Greubel_, Mar 12 2021: (Start)
%F a(n) = 2*A083880(n) + 8*A081182(n).
%F a(n) = (1/2)*Sum_{k=0..n} binomial(n,k)*4^(n-k)*(5*Q(k) + 4*Q(k-1)), where Q(n) = Pell-Lucas(n) = A002203(n). (End)
%t LinearRecurrence[{10,-23},{1,13},20] (* _Harvey P. Dale_, Oct 15 2015 *)
%o (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((1+4*r)*(5+r)^n+(1-4*r)*(5-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Aug 17 2009
%o (PARI) my(x='x+O('x^50)); Vec((1+3*x)/(1-10*x+23*x^2)) \\ _G. C. Greubel_, Sep 13 2017
%o (Sage) [( (1+3*x)/(1-10*x+23*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), this sequence (m=4), A164598 (m=5), A164599 (m=6), A081185 (m=7), A164600 (m=8).
%Y Cf. A081182, A083880, A164587.
%K nonn
%O 0,2
%A Al Hakanson (hawkuu(AT)gmail.com), Aug 12 2009
%E Edited and extended beyond a(5) by _Klaus Brockhaus_, Aug 17 2009
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