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%I #26 Jan 25 2023 09:47:21
%S 1,3,16,90,508,2868,16192,91416,516112,2913840,16450816,92877216,
%T 524361664,2960415552,16713769984,94361788800,532743192832,
%U 3007735579392,16980927090688,95870091385344,541258694130688
%N a(n) = ((3 + sqrt(7))^n + (3 - sqrt(7))^n)/2.
%C Binomial transform of A108851.
%C Inverse binomial transform of A146964.
%H Vincenzo Librandi, <a href="/A146963/b146963.txt">Table of n, a(n) for n = 0..158</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-2).
%F From _Philippe Deléham_ and _Klaus Brockhaus_, Nov 05 2008: (Start)
%F a(n) = 6*a(n-1) - 2*a(n-2) with a(0)=1, a(1)=3.
%F G.f.: (1-3*x)/(1-6*x+2*x^2). (End)
%F a(n) = (Sum_{k=0..n} A098158(n,k)*3^(2*k)*7^(n-k))/3^n. - _Philippe Deléham_, Nov 06 2008
%F E.g.f.: exp(3*x)*cosh(sqrt(7)*x). - _G. C. Greubel_, Jan 08 2020
%F a(n) = A154244(n)-3*A154244(n-1). - _R. J. Mathar_, Jan 25 2023
%p seq(coeff(series((1-3*x)/(1-6*x+2*x^2), x, n+1), x, n), n = 0..25); # _G. C. Greubel_, Jan 08 2020
%t Transpose[NestList[Join[{Last[#],6Last[#]-2First[#]}]&,{1,3},25]] [[1]] (* or *) CoefficientList[Series[(1-3x)/(1-6x+2x^2),{x,0,25}],x] (* _Harvey P. Dale_, Apr 11 2011 *)
%t LinearRecurrence[{6,-2}, {1,3}, 25] (* _G. C. Greubel_, Jan 08 2020 *)
%o (Magma) Z<x>:= PolynomialRing(Integers()); N<r7>:=NumberField(x^2-7); S:=[ ((3+r7)^n+(3-r7)^n)/2: n in [0..25] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Nov 05 2008
%o (PARI) my(x='x+O('x^25)); Vec((1-3*x)/(1-6*x+2*x^2)) \\ _G. C. Greubel_, Jan 08 2020
%o (Sage)
%o def A146963_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( (1-3*x)/(1-6*x+2*x^2) ).list()
%o A146963_list(25) # _G. C. Greubel_, Jan 08 2020
%o (GAP) a:=[1,3];; for n in [3..25] do a[n]:=6*a[n-1]-2*a[n-2]; od; a; # _G. C. Greubel_, Jan 08 2020
%Y Cf. A098158, A108851, A146964.
%K nonn
%O 0,2
%A Al Hakanson (hawkuu(AT)gmail.com), Nov 03 2008
%E Extended beyond a(7) by _Klaus Brockhaus_, Nov 05 2008
%E Edited by _Klaus Brockhaus_, Jul 16 2009
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