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 A049680 #29 Jul 18 2025 01:52:10
%S A049680 2,2,5,11,27,67,170,436,1127,2927,7625,19901,52002,135982,355745,
%T A049680 930931,2436527,6377807,16695530,43706576,114420627,299549527,
%U A049680 784218605,2053091161,5375030402,14071960442,36840786845,96450296411,252509934627,661079236027,1730727334250
%N A049680 a(n) = (L(n) + L(2*n))/2, where L = A000032 (the Lucas sequence).
%H A049680 Indranil Ghosh, <a href="/A049680/b049680.txt">Table of n, a(n) for n = 0..2388</a>
%H A049680 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-3,-2,1).
%F A049680 Binomial transform of trace(A^n)/4, where A is the adjacency matrix of path graph P_4 (A005248 with interpolated zeros). - _Paul Barry_, Apr 24 2004
%F A049680 From _George F. Johnson_, Feb 04 2013: (Start)
%F A049680 G.f.: (1-x)*(2-4*x-x^2)/ ( (1-x-x^2)*(1-3*x+x^2) ).
%F A049680 a(n) = 4*a(n-1) - 3*a(n-2) - 2*a(n-3) + a(n-4) for n>3. (End)
%e A049680 a(8) = (L(8) + L(2 * 8)) / 2 = (47 + 2207) / 2 = 2254 / 2 = 1127. - _Indranil Ghosh_, Feb 06 2017
%t A049680 LinearRecurrence[{4,-3,-2,1},{2,2,5,11},30] (* _Harvey P. Dale_, Nov 22 2015 *)
%t A049680 Table[(LucasL[n] + LucasL[2*n])/2, {n,0,30}] (* _G. C. Greubel_, Dec 02 2017 *)
%o A049680 (PARI) my(x='x+O('x^30)); Vec((1-x)*(2-4*x-x^2)/ ( (1-x-x^2)*(1-3*x+x^2) )) \\ _G. C. Greubel_, Dec 02 2017
%o A049680 (Magma) [(Lucas(n) + Lucas(2*n))/2: n in [0..30]]; // _G. C. Greubel_, Dec 02 2017
%K A049680 nonn,easy
%O A049680 0,1
%A A049680 _Clark Kimberling_