A215493 a(n) = 7*a(n-1) - 14*a(n-2) + 7*a(n-3) with a(0)=0, a(1)=1, a(2)=4.

0, 1, 4, 14, 49, 175, 637, 2352, 8771, 32928, 124166, 469567, 1779141, 6749211, 25623472, 97329337, 369821228, 1405502182, 5342323441, 20307982135, 77201862045, 293497548512, 1115812645899, 4242135876440, 16128056932078, 61317184775679, 233122447515741

Offset: 0

Author

Roman Witula, Aug 13 2012

Keywords

• nonn
• easy

Comments

The Berndt-type sequence number 4 for the argument 2Pi/7 - see also A215007, A215008, A215143 and A215494.

We have a(n)=A079309(n) for n=1..6, and A079309(7)-a(7)=1.

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
• Index entries for linear recurrences with constant coefficients, signature (7,-14,7).

Programs

Magma

I:=[0,1,4]; [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}, {0,1,4}, 50]

PARI

x='x+O('x^30); concat([0], Vec(x*(1-3*x)/(1-7*x+14*x^2-7*x^3))) \\ G. C. Greubel, Apr 23 2018

Formula

a(n)*sqrt(7) = s(1)^(2n-1) + s(2)^(2n-1) + s(4)^(2n-1), where s(j) := 2*Sin(2*Pi*j/7) (for the sums of the respective even powers see A215494, see also A094429, A115146). For the proof of these formula see Witula-Slota's paper.

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

a(n) = A275830(2*n-1)/(7^n). - Kai Wang, May 25 2017