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 A215948 a(n) = 3^n*A(2*n), where A(n) = 3*A(n-1) + A(n-2) - A(n-3)/3 with A(0)=A(1)=3, A(2)=11. 6
 3, 33, 1035, 33273, 1070163, 34420113, 1107069147, 35607149289, 1145248319907, 36835122733569, 1184744167018155, 38105444942752473, 1225602095969542131, 39419576386041628017, 1267869080483024344443, 40779027899804588036553, 1311593714249667872790339 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS The Berndt-type sequence number 12 for the argument 2*Pi/9 defined by the first trigonometric relations from the section "Formula" below (it is the complement of the sequence A215945). For more information see comments to A215945. We note that all a(n)/3 and 3^(-1 + floor((n+3)/3))*A(n) = A216034(n) are integers. REFERENCES D. Chmiela and R. Witula, Two parametric quasi-Fibonacci numbers of the ninth order, (submitted, 2012). R. Witula, Ramanujan type formulas for arguments 2Pi/7 and 2Pi/9, Demonstratio Math. (in press, 2012). LINKS Roman Witula and Damian Slota, New Ramanujan-Type Formulas and Quasi-Fibonacci Numbers of Order 7, Journal of Integer Sequences, Vol. 10 (2007), Article 07.5.6. Index entries for linear recurrences with constant coefficients, signature (33,-27,3). FORMULA a(n) = t(1)^(2*n) + t(2)^(2*n) + t(4)^(2*n) = (-sqrt(3) + 4*s(1))^(2*n) + (sqrt(3) + 4*s(2))^(2*n) + (-sqrt(3) + 4*s(4))^(2*n), where t(j) := tan(2*Pi*j/9) and s(j) := sin(2*Pi*j/9). For the respective sums of odd powers - see A215945. a(n) = 33*a(n-1) - 27*a(n-2) + 3*a(n-3). G.f.: 3*(1-22*x+9*x^2)/(1-33*x+27*x^2-3*x^3). a(n) = cot(Pi/18)^(2*n) + cot(5*Pi/18)^(2*n) + cot(7*Pi/18)^(2*n). - Greg Dresden, Oct 01 2020 EXAMPLE We have t(1)^4 + t(2)^4 + t(4)^4 = 1035 = (345/11)*(t(1)^2 + t(2)^2 + t(4)^2) and (1 - 4*s(1)/sqrt(3))^4 + (1 + 4*s(2)/sqrt(3))^4 + (1 - 4*s(4)/sqrt(3))^4 = 115. Moreover we get a(2)/a(1) = 31,(36), a(3)/a(1) = 1008,(27), a(4)/a(1) = 32429,(18). MATHEMATICA LinearRecurrence[{33, -27, 3}, {3, 33, 1035}, 50] CROSSREFS Cf. A215945, A216034, A215829, A215794, A215575. Sequence in context: A086894 A255930 A255883 * A012487 A188387 A113111 Adjacent sequences:  A215945 A215946 A215947 * A215949 A215950 A215951 KEYWORD nonn,easy AUTHOR Roman Witula, Aug 28 2012 STATUS approved

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Last modified November 30 08:19 EST 2021. Contains 349419 sequences. (Running on oeis4.)