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.

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

Offset: 0

Author

Roman Witula, Aug 14 2012

Keywords

• nonn
• easy

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.

Examples

			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.
		

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).

Programs

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

Mathematica

LinearRecurrence[{7,-14,7}, {0,7,35}, 50]

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

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