0,3

This is the Berndt-type sequence number 18 for the argument 2*Pi/7 defined by the relation

a(n)*sqrt(7) = c(4)*s(1)^(2n+1) + c(2)*s(4)^(2n+1) + c(1)*s(2)^(2n+1) = (1/s(4))*s(1)^(2n+2) + (1/s(2))*s(4)^(2n+2) + (1/s(1))*s(2)^(2n+2), where c(j) := 2*cos(2*Pi*j/7) and s(j) := 2*sin(2*Pi*j/7) (for the sums of the respective even powers see A094429). For the proof of this formula see the Witula/Slota and Witula references.

The definitions of the other Berndt-type sequences for the argument 2*Pi/7 (with numbers from 1 to 17) are in the cross references.

We note that all numbers of the form a(n)*7^(-floor((n+1)/3)) = A217444(n) are integers.

It can be proved that Sum_{k=2..n}a(k) = 7*(a(n-1) - a(n-2)).

G. C. Greubel, Table of n, a(n) for n = 0..1000

R. Witula, Ramanujan type trigonometric formulas, Demonstratio Math., Vol. XLV, No. 4, 2012, pp. 789-796.

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).

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

Writing c(j) as cj and s(k) as sk,

we have 7*sqrt(7) = c4*s1^5 + c2*s4^5 + c1*s2^5

and c4*s1^13 + c2*s4^13 + c1*s2^13 = 4(c4*s1^11 + c2*s4^11 + c1*s2^11).

We note that a(9) = 87*a(3)*a(2)^2 and a(11) = 2*a(3)*a(5)*a(2)^2.

LinearRecurrence[{7, -14, 7}, {0, 1, 7}, 30]

CoefficientList[Series[x/(1 - 7*x + 14*x^2 - 7*x^3), {x, 0, 50}], x] (* G. C. Greubel, Apr 16 2017 *)

(Maxima)

a[0]:0$

a[1]:1$

a[2]:7$

a[n]:=7*a[n-1] - 14*a[n-2] + 7*a[n-3];

makelist(a[n], n, 0, 25); /* Martin Ettl, Oct 11 2012 */

(PARI) concat(0, Vec(x/(1-7*x+14*x^2-7*x^3) + O(x^40))) \\ Michel Marcus, Jul 25 2015

(Magma) I:=[0, 1, 7]; [n le 3 select I[n] else 7*Self(n-1)-14*Self(n-2)+7*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jul 26 2015

Cf. A033304, A094429, A094430, A094648, A108716, A215007, A215008, A215143, A215493, A215494, A215510, A215512, A215575, A215694, A215695, A215794, A215817, A215828, A215877, A217444.

Sequence in context: A022635 A336602 A370391 * A000588 A005285 A371964

Adjacent sequences: A217271 A217272 A217273 * A217275 A217276 A217277

nonn,easy

Roman Witula, Sep 29 2012

approved