A084157 - OEIS

%I #17 Oct 13 2022 13:02:51

%S 0,1,4,22,112,556,2704,13000,62080,295312,1401664,6644320,31472896,

%T 149017792,705395968,3338614912,15800258560,74772443392,353840161792,

%U 1674425579008,7923565146112,37494981225472,177428889407488

%N 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 Binomial transform of A084156.

%H G. C. Greubel, <a href="/A084157/b084157.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-16,0,12).

%F a(n) = (A083881(n) - A026150(n))/2.

%F a(n) = 8*a(n-1) - 16*a(n-2) + 12*a(n-4).

%F a(n) = ((3+sqrt(3))^n + (3-sqrt(3))^n - (1+sqrt(3))^n - (1-sqrt(3))^n)/4.

%F G.f.: x*(1-4*x+6*x^2)/((1-2*x-2*x^2)*(1-6*x+6*x^2)).

%F E.g.f.: exp(2*x)*sinh(x)*cosh(sqrt(3)*x).

%F From _G. C. Greubel_, Oct 11 2022: (Start)

%F a(2*n) = A003462(n)*A026150(2*n) = A003462(n)*A080040(2*n)/2.

%F a(2*n+1) = (1/2)*(3^(n+1)*A002605(2*n+1) - A026150(2*n+1)). (End)

%t LinearRecurrence[{8,-16,0,12},{0,1,4,22},30] (* _Harvey P. Dale_, Feb 19 2017 *)

%o (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 (SageMath)

%o A083881 = BinaryRecurrenceSequence(6,-6,1,3)

%o A026150 = BinaryRecurrenceSequence(2,2,1,1)

%o def A084157(n): return (A083881(n) - A026150(n))/2

%o [A084157(n) for n in range(41)] # _G. C. Greubel_, Oct 11 2022

%Y Cf. A026150, A083881, A084155, A084156.

%Y Cf. A002605, A003462, A080040.

%K easy,nonn

%O 0,3

%A _Paul Barry_, May 16 2003