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 A215828 a(n) = 7^(floor(n/3))*A(n), where A(n) = A(n-1) + A(n-2) + A(n-3)/7, with A(0)=3, A(1)=1, A(2)=3. 10
 3, 1, 3, 31, 53, 87, 1011, 1673, 2771, 32119, 53189, 88079, 1020995, 1690737, 2799811, 32454831, 53744245, 88998887, 1031656755, 1708393209, 2829048851, 32793751175, 54305486341, 89928286367, 1042430160131, 1726233651041, 2858592097539, 33136210400191 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS The Berndt-type sequence number 13 for the argument 2Pi/7   defined by the relation ((-sqrt(7))^n)*A(n) = t(1)^n + t(2)^n + t(4)^n = (-sqrt(7) + 4*s(1))^n + (-sqrt(7) + 4*s(2))^n + (-sqrt(7) + 4*s(4))^n, where t(j) := tan(2*Pi*j/7) and s(j) := sin(2*Pi*j/7), and the fact that all numbers 7^(floor(n/3))*A(n) are integers. We note that ((-sqrt(7))^n)*A(n) = B(n), where B(n) is defined in the comments to A215575. For more details see also A108716, A215794, Witula-Slota's (Section 6) and Witula's (Remark 11) papers. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 Roman Witula, Ramanujan Type Trigonometric Formulas: The General Form for the Argument 2*Pi/7, Journal of Integer Sequences, Vol. 12 (2009), Article 09.8.5. R. Witula, P. Lorenc, M. Rozanski, M. Szweda, Sums of the rational powers of roots of cubic polynomials, Zeszyty Naukowe Politechniki Slaskiej, Seria: Matematyka Stosowana z. 4, Nr. kol. 1920, 2014. 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 (0,0,31,0,0,25,0,0,1). FORMULA G.f.: (x^8-5*x^7+25*x^6+6*x^5-22*x^4+62*x^3-3*x^2-x-3)/(x^9+25*x^6+31*x^3-1). [Colin Barker, Oct 28 2012] EXAMPLE We have A(3)=31/7, A(4)=53/7 and A(5)=87/7. On the other hand we have a(2)+a(3)+a(4)=a(5). MATHEMATICA CoefficientList[Series[(x^8 - 5 x^7 + 25 x^6 + 6 x^5 - 22 x^4 + 62 x^3 - 3 x^2 - x - 3)/(x^9 + 25 x^6 + 31 x^3 - 1), {x, 0, 30}], x] (* Vincenzo Librandi, Mar 19 2013 *) PROG (MAGMA) /* By definition: */ i:=28; I:=[3, 1, 3]; A:=[m le 3 select I[m] else Self(m-1)+Self(m-2)+Self(m-3)/7: m in [1..i]]; [7^(Floor((n-1)/3))*A[n]: n in [1..i]]; // Bruno Berselli, Oct 28 2012 CROSSREFS Cf. A215575, A108716, A215794. Sequence in context: A320952 A128777 A286892 * A067009 A229755 A257634 Adjacent sequences:  A215825 A215826 A215827 * A215829 A215830 A215831 KEYWORD nonn,easy AUTHOR Roman Witula, Aug 24 2012 STATUS approved

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Last modified February 28 06:55 EST 2020. Contains 332321 sequences. (Running on oeis4.)