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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
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
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.
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
KEYWORD
nonn,easy
AUTHOR
Roman Witula, Aug 24 2012
STATUS
approved