A215143 a(n) = 7*a(n-1) -14*a(n-2) +7*a(n-3), with a(0)=1, a(1)=2, a(2)=7.

1, 2, 7, 28, 112, 441, 1715, 6615, 25382, 97069, 370440, 1411788, 5375839, 20458921, 77833217, 296038498, 1125816895, 4281011812, 16277915640, 61891962377, 235320000363, 894697938743, 3401649302758, 12933013979445, 49170893188704, 186945601728004, 710757805310287

Offset: 0

Author

Roman Witula, Aug 04 2012

Keywords

• nonn
• easy

Comments

The Berndt-type sequence number 3 for the argument 2Pi/7 (see A215007 and A215008 for the respective sequences numbers 1 and 2) is defined by the following relations: sqrt(7) *a(n) = s(1)*s(2)^(2n) + s(2)*s(4)^(2n) + s(4)*s(1)^(2n) = s(4)*s(1)^(2n) + s(1)*s(2)^(2n) + s(2)*s(4)^(2n), where s(j) := 2*sin(2*Pi*j/7).

References

• R. Witula, Complex numbers, Polynomials and Fractial Partial Decompositions, T.3, Silesian Technical University Press, Gliwice 2010 (in Polish).

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, Ramanujan Type Trigonometric Formulae, Demonstratio Math. 45 (2012) 779-796.
• 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 (7,-14,7).

Crossrefs

Cf. A215007, A215008.

Programs

Magma

I:=[1,2,7]; [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 19 2018

Mathematica

LinearRecurrence[{7,-14,7},{1,2,7},40]

PARI

Vec((1-5*x+7*x^2)/(1-7*x+14*x^2-7*x^3) + O(x^30)) \\ Michel Marcus, Apr 19 2016

Formula

G.f.: (1-5*x+7*x^2)/(1-7*x+14*x^2-7*x^3).