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A323210 a(n) = 9*J(n)^2 where J(n) are the Jacobsthal numbers A001045 with J(0) = 1.

%I A323210 #25 Mar 06 2022 10:04:27
%S A323210 1,9,9,81,225,1089,3969,16641,65025,263169,1046529,4198401,16769025,
%T A323210 67125249,268402689,1073807361,4294836225,17180131329,68718952449,
%U A323210 274878955521,1099509530625,4398050705409,17592177655809,70368760954881,281474943156225,1125899973951489
%N A323210 a(n) = 9*J(n)^2 where J(n) are the Jacobsthal numbers A001045 with J(0) = 1.
%C A323210 _Colin Barker_ conjectures that A208556 is a shifted version of this sequence.
%H A323210 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,6,-8).
%F A323210 a(n) = Product_{k=1..n} T(n, k) where T(n, k) = t(n,k)*conjugate(t(n,k)) and t(n,k) = 3*cos(Pi*k/n) - i*sin(Pi*k/n), i is the imaginary unit.
%F A323210 a(n) = [x^n] (8*x^3 - 24*x^2 + 6*x + 1)/((4*x - 1)*(2*x + 1)*(x - 1)).
%F A323210 a(n) = n! [x^n] (1 + exp(x) - 2*exp(-2*x) + exp(4*x)).
%F A323210 a(n) = 3*a(n-1) + 6*a(n-2) - 8*a(n-3) for n >= 4.
%F A323210 A062510(n) = sqrt(a(n)) for n > 0.
%F A323210 a(n) = 4^n-2*(-2)^n+1, n>0. - _R. J. Mathar_, Mar 06 2022
%p A323210 gf := (8*x^3 - 24*x^2 + 6*x + 1)/((4*x - 1)*(2*x + 1)*(x - 1)):
%p A323210 ser := series(gf,x,32): seq(coeff(ser,x,n), n=0..25);
%t A323210 LinearRecurrence[{3, 6, -8}, {1, 9, 9, 81}, 25]
%o A323210 (Sage) # Demonstrates the product formula.
%o A323210 CC = ComplexField(200)
%o A323210 def t(n,k): return CC(3)*cos(CC(pi*k/n)) - CC(i)*sin(CC(pi*k/n))
%o A323210 def T(n,k): return t(n,k)*(t(n,k).conjugate())
%o A323210 def a(n): return prod(T(n,k) for k in (1..n))
%o A323210 print([a(n).real().round() for n in (0..29)])
%Y A323210 Cf. A001045, A062510, A139818, A208556, A322942, A323232.
%K A323210 nonn,easy
%O A323210 0,2
%A A323210 _Peter Luschny_, Jan 09 2019