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%I #38 Oct 21 2022 22:09:58
%S 1,10,38,132,452,1544,5272,18000,61456,209824,716384,2445888,8350784,
%T 28511360,97343872,332352768,1134723328,3874187776,13227304448,
%U 45160842240,154188760064,526433355776,1797355902976,6136556900352
%N a(n) = ((1+4*sqrt(2))*(2+sqrt(2))^n + (1-4*sqrt(2))*(2-sqrt(2))^n)/2.
%C Binomial transform of A048696. Second binomial transform of A164587. Inverse binomial transform of A164299.
%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="/A164298/b164298.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-2).
%F a(n) = 4*a(n-1) - 2*a(n-2) for n > 1; a(0)=1, a(1)=10.
%F G.f.: (1+6*x)/(1-4*x+2*x^2).
%F E.g.f.: (cosh(sqrt(2)*x) + 4*sqrt(2)*sinh(sqrt(2)*x))*exp(2*x). - _G. C. Greubel_, Sep 12 2017
%F From _G. C. Greubel_, Mar 12 2021: (Start)
%F a(n) = A056236(n) + 8*A007070(n-1).
%F a(n) = (1/2)*Sum_{k=0..n} binomial(n,k)*(5*Q(k) + 4*Q(k-1)), where Q(n) = Pell-Lucas(n) = A002203(n). (End)
%p a:=n->((1+4*sqrt(2))*(2+sqrt(2))^n+(1-4*sqrt(2))*(2-sqrt(2))^n)/2: seq(floor(a(n)),n=0..25); # _Muniru A Asiru_, Dec 15 2018
%t LinearRecurrence[{4,-2}, {1,10}, 50] (* or *) CoefficientList[Series[(1 + 6*x)/(1 - 4*x + 2*x^2), {x,0,50}], x] (* _G. C. Greubel_, Sep 12 2017 *)
%o (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((1+4*r)*(2+r)^n+(1-4*r)*(2-r)^n)/2: n in [0..27] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Aug 17 2009
%o (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1+6*x)/(1-4*x+2*x^2) )); // _G. C. Greubel_, Dec 14 2018
%o (PARI) my(x='x+O('x^50)); Vec((1+6*x)/(1-4*x+2*x^2)) \\ _G. C. Greubel_, Sep 12 2017
%o (Sage) [( (1+6*x)/(1-4*x+2*x^2) ).series(x,n+1).list()[n] for n in (0..30)] # _G. C. Greubel_, Dec 14 2018; Mar 12, 2021
%Y Sequences in the class a(n, m): this sequence (m=1), A164299 (m=2), A164300 (m=3), A164301 (m=4), A164598 (m=5), A164599 (m=6), A081185 (m=7), A164600 (m=8).
%Y Cf. A007070, A016921, A048696, A056236, A112032, A164587, A241204.
%Y Cf. A016116(n+1).
%K nonn,easy
%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