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A084157 a(n) = 8*a(n-1) - 16*a(n-2) + 12*a(n-4) with a(0)=0, a(1)=1, a(2)=4, a(3)=22.

%I A084157 #17 Oct 13 2022 13:02:51
%S A084157 0,1,4,22,112,556,2704,13000,62080,295312,1401664,6644320,31472896,
%T A084157 149017792,705395968,3338614912,15800258560,74772443392,353840161792,
%U A084157 1674425579008,7923565146112,37494981225472,177428889407488
%N A084157 a(n) = 8*a(n-1) - 16*a(n-2) + 12*a(n-4) with a(0)=0, a(1)=1, a(2)=4, a(3)=22.
%C A084157 Binomial transform of A084156.
%H A084157 G. C. Greubel, <a href="/A084157/b084157.txt">Table of n, a(n) for n = 0..1000</a>
%H A084157 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-16,0,12).
%F A084157 a(n) = (A083881(n) - A026150(n))/2.
%F A084157 a(n) = 8*a(n-1) - 16*a(n-2) + 12*a(n-4).
%F A084157 a(n) = ((3+sqrt(3))^n + (3-sqrt(3))^n - (1+sqrt(3))^n - (1-sqrt(3))^n)/4.
%F A084157 G.f.: x*(1-4*x+6*x^2)/((1-2*x-2*x^2)*(1-6*x+6*x^2)).
%F A084157 E.g.f.: exp(2*x)*sinh(x)*cosh(sqrt(3)*x).
%F A084157 From _G. C. Greubel_, Oct 11 2022: (Start)
%F A084157 a(2*n) = A003462(n)*A026150(2*n) = A003462(n)*A080040(2*n)/2.
%F A084157 a(2*n+1) = (1/2)*(3^(n+1)*A002605(2*n+1) - A026150(2*n+1)). (End)
%t A084157 LinearRecurrence[{8,-16,0,12},{0,1,4,22},30] (* _Harvey P. Dale_, Feb 19 2017 *)
%o A084157 (Magma) I:=[0,1,4,22]; [n le 4 select I[n] else 8*Self(n-1) -16*Self(n-2) +12*Self(n-4): n in [1..41]]; // _G. C. Greubel_, Oct 11 2022
%o A084157 (SageMath)
%o A084157 A083881 = BinaryRecurrenceSequence(6,-6,1,3)
%o A084157 A026150 = BinaryRecurrenceSequence(2,2,1,1)
%o A084157 def A084157(n): return (A083881(n) - A026150(n))/2
%o A084157 [A084157(n) for n in range(41)] # _G. C. Greubel_, Oct 11 2022
%Y A084157 Cf. A026150, A083881, A084155, A084156.
%Y A084157 Cf. A002605, A003462, A080040.
%K A084157 easy,nonn
%O A084157 0,3
%A A084157 _Paul Barry_, May 16 2003