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A084130 a(n) = 8*a(n-1) - 8*a(n-2), a(0)=1, a(1)=4.

%I A084130 #43 Apr 13 2025 19:30:20
%S A084130 1,4,24,160,1088,7424,50688,346112,2363392,16138240,110198784,
%T A084130 752484352,5138284544,35086401536,239584935936,1635988275200,
%U A084130 11171226714112,76281907511296,520885446377472,3556828310929408,24287542916415488
%N A084130 a(n) = 8*a(n-1) - 8*a(n-2), a(0)=1, a(1)=4.
%C A084130 Binomial transform of A001541.
%C A084130 Let A be the unit-primitive matrix (see [Jeffery]) A = A_(8,3) = [0,0,0,1; 0,0,2,0; 0,2,0,1; 2,0,2,0]. Then A084130(n) = (1/4)*Trace(A^(2*n)). (Cf. A006012, A001333.) - _L. Edson Jeffery_, Apr 04 2011
%C A084130 a(n) is also the rational part of the Q(sqrt(2)) integer giving the length L(n) of a variant of the Lévy C-curve, given by _Kival Ngaokrajang_, at iteration step n. See A057084. - _Wolfdieter Lang_, Dec 18 2014
%H A084130 G. C. Greubel, <a href="/A084130/b084130.txt">Table of n, a(n) for n = 0..1000</a>
%H A084130 L. E. Jeffery, <a href="/wiki/User:L._Edson_Jeffery/Unit-Primitive_Matrices">Unit-primitive matrices</a>
%H A084130 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,-8).
%F A084130 a(n) = (4+sqrt(8))^n/2 + (4-sqrt(8))^n/2.
%F A084130 G.f.: (1-4*x)/(1-8*x+8*x^2).
%F A084130 E.g.f.: exp(4*x)*cosh(sqrt(8)*x).
%F A084130 a(n) = A057084(n) - 4*A057084(n-1). - _R. J. Mathar_, Nov 10 2013
%F A084130 From _G. C. Greubel_, Oct 13 2022: (Start)
%F A084130 a(2*n) = 2^(3*n-1)*A002203(2*n).
%F A084130 a(2*n+1) = 2^(3*n+2)*A000129(2*n+1). (End)
%t A084130 LinearRecurrence[{8,-8},{1,4},30] (* _Harvey P. Dale_, Sep 25 2014 *)
%o A084130 (PARI) {a(n)= if(n<0, 0, real((4+ 2*quadgen(8))^n))}
%o A084130 (Magma) [n le 2 select 4^(n-1) else 8*(Self(n-1) -Self(n-2)): n in [1..41]]; // _G. C. Greubel_, Oct 13 2022
%o A084130 (SageMath)
%o A084130 A084130=BinaryRecurrenceSequence(8,-8,1,4)
%o A084130 [A084130(n) for n in range(41)] # _G. C. Greubel_, Oct 13 2022
%Y A084130 Cf. A000129, A001333, A001541, A002203, A006012, A057084, A084130, A084131.
%K A084130 easy,nonn
%O A084130 0,2
%A A084130 _Paul Barry_, May 16 2003