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 A323210 a(n) = 9*J(n)^2 where J(n) are the Jacobsthal numbers A001045 with J(0) = 1. 0
 1, 9, 9, 81, 225, 1089, 3969, 16641, 65025, 263169, 1046529, 4198401, 16769025, 67125249, 268402689, 1073807361, 4294836225, 17180131329, 68718952449, 274878955521, 1099509530625, 4398050705409, 17592177655809, 70368760954881, 281474943156225, 1125899973951489 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Colin Barker conjectures that A208556 is a shifted version of this sequence. LINKS Table of n, a(n) for n=0..25. Index entries for linear recurrences with constant coefficients, signature (3,6,-8). FORMULA 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. a(n) = [x^n] (8*x^3 - 24*x^2 + 6*x + 1)/((4*x - 1)*(2*x + 1)*(x - 1)). a(n) = n! [x^n] (1 + exp(x) - 2*exp(-2*x) + exp(4*x)). a(n) = 3*a(n-1) + 6*a(n-2) - 8*a(n-3) for n >= 4. A062510(n) = sqrt(a(n)) for n > 0. a(n) = 4^n-2*(-2)^n+1, n>0. - R. J. Mathar, Mar 06 2022 MAPLE gf := (8*x^3 - 24*x^2 + 6*x + 1)/((4*x - 1)*(2*x + 1)*(x - 1)): ser := series(gf, x, 32): seq(coeff(ser, x, n), n=0..25); MATHEMATICA LinearRecurrence[{3, 6, -8}, {1, 9, 9, 81}, 25] PROG (Sage) # Demonstrates the product formula. CC = ComplexField(200) def t(n, k): return CC(3)*cos(CC(pi*k/n)) - CC(i)*sin(CC(pi*k/n)) def T(n, k): return t(n, k)*(t(n, k).conjugate()) def a(n): return prod(T(n, k) for k in (1..n)) print([a(n).real().round() for n in (0..29)]) CROSSREFS Cf. A001045, A062510, A139818, A208556, A322942, A323232. Sequence in context: A223744 A112296 A038299 * A215272 A165427 A050683 Adjacent sequences: A323207 A323208 A323209 * A323211 A323212 A323213 KEYWORD nonn,easy AUTHOR Peter Luschny, Jan 09 2019 STATUS approved

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Last modified August 9 16:51 EDT 2024. Contains 375044 sequences. (Running on oeis4.)