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%I #30 Sep 08 2022 08:46:08
%S 1,1,7,19,91,331,1387,5419,21931,87211,349867,1397419,5593771,
%T 22366891,89483947,357903019,1431677611,5726579371,22906579627,
%U 91625794219,366504225451,1466014804651,5864063412907,23456245263019,93824997829291,375299957762731
%N a(n) = (1^n + (-2)^n + 4^n)/3.
%H Vincenzo Librandi, <a href="/A245489/b245489.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,6,-8).
%F G.f.: (1 - 2*x - 2*x^2) / ((1 - x) * (1 + 2*x) * (1 - 4*x)).
%F 0 = 8*a(n) - 6*a(n+1) - 3*a(n+2) + a(n+3) for all n in Z.
%F a(2*n) = A018240(4*n + 3). a(2*n + 1) = A129362(4*n).
%F a(n) = A001045(3*n)/(3*A001045(n)) for n >= 1. - _Peter Bala_, Apr 06 2015
%F E.g.f.: (exp(x) + exp(4*x) + exp(-2*x))/3. - _G. C. Greubel_, Sep 21 2019
%e G.f. = 1 + x + 7*x^2 + 19*x^3 + 91*x^4 + 331*x^5 + 1387*x^6 + 5419*x^7 + ...
%p seq((1 +(-2)^n +4^n)/3, n=0..30); # _G. C. Greubel_, Sep 21 2019
%t CoefficientList[Series[(1-2x-2x^2)/((1-x)(1+2x)(1-4x)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Jul 25 2014 *)
%t LinearRecurrence[{3,6,-8},{1,1,7},30] (* _Harvey P. Dale_, Dec 04 2018 *)
%o (PARI) {a(n) = (1^n + (-2)^n + 4^n) / 3};
%o (PARI) {a(n) = if( n<0, 4^n, 1) * polcoeff( (1 - 2*x - 2*x^2) / ((1 - x) * (1 + 2*x) * (1 - 4*x)) + x * O(x^abs(n)), abs(n))};
%o (Magma) [(1^n + (-2)^n + 4^n) / 3 : n in [0..30]]; // _Vincenzo Librandi_, Jul 25 2014
%o (Sage) [(1 +(-2)^n +4^n)/3 for n in (0..30)] # _G. C. Greubel_, Sep 21 2019
%o (GAP) List([0..30], n-> (1 +(-2)^n +4^n)/3); # _G. C. Greubel_, Sep 21 2019
%Y Cf. A001045, A018240, A129362.
%K nonn,easy
%O 0,3
%A _Michael Somos_, Jul 23 2014
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