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 A215510 a(n) = 7*a(n-1) - 14*a(n-2) + 7*a(n-3) with a(0)=0, a(1)=7, a(2)=35. 15
 0, 7, 35, 147, 588, 2303, 8918, 34300, 131369, 501809, 1913597, 7289436, 27748357, 105581574, 401620072, 1527436967, 5808448779, 22086364419, 83978326796, 319298327159, 1213996265902, 4615645568660, 17548659548105, 66719552736809, 253665154464813 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The Berndt-type sequence number 6 for the argument 2Pi/7 (see A215007, A215008, A215143, A215493 and A215494 for the respective sequences numbers 1-5) is defined by the following relation: a(n) = s(1)*s(2)^(2n+1) + s(2)*s(4)^(2n+1) + s(4)*s(1)^(2n+1), where s(j) := 2*sin(2*Pi*j/7). For the respective sums with even powers see A215143. We note that a(4)=49*sqrt(7)*(s(1)*s(4)^(-6) + s(2)*s(4)^(-6) + s(4)*s(1)^(-6)) - see the respective value of the sequence y*(n) in Witula-Slota's paper. LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 B. C. Berndt, A. Zaharescu, Finite trigonometric sums and class numbers, Math. Ann. 330 (2004), 551-575. B. C. Berndt, L.-C. Zhang, Ramanujan's identities for eta-functions, Math. Ann. 292 (1992), 561-573. Z.-G. Liu, Some Eisenstein series identities related to modular equations of the seventh order, Pacific J. Math. 209 (2003), 103-130. 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. Roman Witula, Ramanujan Type Trigonometric Formulae, Demonstratio Math. 45 (2012) 779-796. Index entries for linear recurrences with constant coefficients, signature (7,-14,7). FORMULA G.f.: (7*x-14*x^2)/(1-7*x+14*x^2-7*x^3). a(n) = 7*A215008(n). - R. J. Mathar, Nov 07 2015 EXAMPLE We have (1-7*x+14*x^2-7*x^3)*(a(1)*x + a(3)*x^2 + a(5)*x^3 + ...) = b(1)*x - b(2)*x^2 + b(3)*x^3 - b(4)*x^4 + (b(5)-2b(2))*x^5 + ..., where b(n)=A094430(n) for n=1,...,5. MATHEMATICA LinearRecurrence[{7, -14, 7}, {0, 7, 35}, 50] PROG (PARI) x='x+O('x^30); concat([0], Vec((7*x-14*x^2)/(1-7*x+14*x^2-7*x^3))) \\ G. C. Greubel, Apr 23 2018 (Magma) I:=[0, 7, 35]; [n le 3 select I[n] else 7*Self(n-1) - 14*Self(n-2) + 7*Self(n-3): n in [1..30]]; // G. C. Greubel, Apr 23 2018 CROSSREFS Sequence in context: A211843 A121163 A223621 * A240423 A094825 A022635 Adjacent sequences: A215507 A215508 A215509 * A215511 A215512 A215513 KEYWORD nonn,easy AUTHOR Roman Witula, Aug 14 2012 STATUS approved

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Last modified November 30 18:14 EST 2023. Contains 367461 sequences. (Running on oeis4.)