This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A164299 #19 Sep 08 2022 08:45:47
%S A164299 1,11,59,277,1249,5555,24587,108637,479713,2117819,9348923,41268805,
%T A164299 182170369,804140579,3549650891,15668921293,69165971521,305313380075,
%U A164299 1347718479803,5949117218293,26260673951137,115920223178771
%N A164299 a(n) = ((1+4*sqrt(2))*(3+sqrt(2))^n + (1-4*sqrt(2))*(3-sqrt(2))^n)/2.
%C A164299 Binomial transform of A164298. Third binomial transform of A164587. Inverse binomial transform of A164300.
%C A164299 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 A164299 G. C. Greubel, <a href="/A164299/b164299.txt">Table of n, a(n) for n = 0..1000</a>
%H A164299 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-7).
%F A164299 a(n) = 6*a(n-1) - 7*a(n-2) for n > 1; a(0) = 1, a(1) = 11.
%F A164299 G.f.: (1+5*x)/(1-6*x+7*x^2).
%F A164299 E.g.f.: (cosh(sqrt(2)*x) + 4*sqrt(2)*sinh(sqrt(2)*x))*exp(3*x). - _G. C. Greubel_, Sep 12 2017
%F A164299 From _G. C. Greubel_, Mar 12 2021: (Start)
%F A164299 a(n) = 2*A083878(n) + 8*A081179(n).
%F A164299 a(n) = (1/2)*Sum_{k=0..n} binomial(n,k)*2^(n-k)*(5*Q(k) + 4*Q(k-1)), where Q(n) = Pell-Lucas(n) = A002203(n). (End)
%t A164299 LinearRecurrence[{6,-7}, {1,11}, 50] (* or *) CoefficientList[Series[(1 + 5*x)/(1 - 6*x + 7*x^2), {x,0,50}], x] (* _G. C. Greubel_, Sep 12 2017 *)
%o A164299 (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((1+4*r)*(3+r)^n+(1-4*r)*(3-r)^n)/2: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Aug 17 2009
%o A164299 (PARI) my(x='x+O('x^50)); Vec((1+5*x)/(1-6*x+7*x^2)) \\ _G. C. Greubel_, Sep 12 2017
%o A164299 (Sage) [( (1+5*x)/(1-6*x+7*x^2) ).series(x,n+1).list()[n] for n in (0..30)] # _G. C. Greubel_, Mar 12 2021
%Y A164299 Sequences in the class a(n, m): A164298 (m=1), this sequence (m=2), A164300 (m=3), A164301 (m=4), A164598 (m=5), A164599 (m=6), A081185 (m=7), A164600 (m=8).
%Y A164299 Cf. A081179, A083878, A164587.
%K A164299 nonn
%O A164299 0,2
%A A164299 Al Hakanson (hawkuu(AT)gmail.com), Aug 12 2009
%E A164299 Edited and extended beyond a(5) by _Klaus Brockhaus_, Aug 17 2009