A164300 - OEIS

%I #29 Sep 08 2022 08:45:47

%S 1,12,82,488,2756,15216,83144,452128,2453008,13294272,72012064,

%T 389976704,2111644736,11433484032,61904845952,335169991168,

%U 1814692086016,9825156811776,53195565289984,288012326955008

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

%C Binomial transform of A164299. Fourth binomial transform of A164587. Inverse binomial transform of A164301.

%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="/A164300/b164300.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 (8,-14).

%F a(n) = 8*a(n-1) - 14*a(n-2) for n > 1; a(0) = 1, a(1) = 12.

%F G.f.: (1+4*x)/(1-8*x+14*x^2).

%F E.g.f.: (4*sqrt(2)*sinh(sqrt(2)*x) + cosh(sqrt(2)*x))*exp(4*x). - _Ilya Gutkovskiy_, Jun 24 2016

%F From _G. C. Greubel_, Mar 12 2021: (Start)

%F a(n) = 2*A083879(n) + 8*A081180(n).

%F a(n) = (1/2)*Sum_{k=0..n} binomial(n,k)*3^(n-k)*(5*Q(k) + 4*Q(k-1)), where Q(n) = Pell-Lucas(n) = A002203(n). (End)

%t LinearRecurrence[{8,-14},{1,12},30] (* _Harvey P. Dale_, Apr 13 2012 *)

%o (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((1+4*r)*(4+r)^n+(1-4*r)*(4-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+4*x)/(1-8*x+14*x^2)) \\ _G. C. Greubel_, Sep 13 2017

%o (Sage) [( (1+4*x)/(1-8*x+14*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), this sequence (m=3), A164301 (m=4), A164598 (m=5), A164599 (m=6), A081185 (m=7), A164600 (m=8).

%Y Cf. A081180, A083879, A164587.

%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