

A215493


a(n) = 7*a(n1)  14*a(n2) + 7*a(n3) with a(0)=0, a(1)=1, a(2)=4.


17



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
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OFFSET

0,3


COMMENTS

The Berndttype 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), 551575.
B. C. Berndt, L.C. Zhang, Ramanujan's identities for etafunctions, Math. Ann. 292 (1992), 561573.
Z.G. Liu, Some Eisenstein series identities related to modular equations of the seventh order, Pacific J. Math. 209 (2003), 103130.
Roman Witula and Damian Slota, New RamanujanType Formulas and QuasiFibonacci 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).


FORMULA

a(n)*sqrt(7) = s(1)^(2n1) + s(2)^(2n1) + s(4)^(2n1), 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 WitulaSlota's paper.
G.f.: x*(13*x)/(17*x+14*x^27*x^3).
a(n) = A275830(2*n1)/(7^n).  Kai Wang, May 25 2017


MATHEMATICA

LinearRecurrence[{7, 14, 7}, {0, 1, 4}, 50]


PROG

(PARI) x='x+O('x^30); concat([0], Vec(x*(13*x)/(17*x+14*x^27*x^3))) \\ G. C. Greubel, Apr 23 2018
(MAGMA) I:=[0, 1, 4]; [n le 3 select I[n] else 7*Self(n1)  14*Self(n2) +7*Self(n3): n in [1..30]]; // G. C. Greubel, Apr 23 2018


CROSSREFS

Sequence in context: A316974 A278026 A001894 * A079309 A026630 A034459
Adjacent sequences: A215490 A215491 A215492 * A215494 A215495 A215496


KEYWORD

nonn,easy


AUTHOR

Roman Witula, Aug 13 2012


STATUS

approved



