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a(n) = (2*sqrt(7)*sin(Pi/7))^n + (-2*sqrt(7)*sin(2*Pi/7))^n + (-2*sqrt(7)*sin(4*Pi/7))^n.
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%I #39 May 26 2017 03:21:44

%S 3,-7,49,-196,1029,-4802,24010,-117649,588245,-2941225,14823774,

%T -74942413,380476866,-1936973136,9886633715,-50563069571,259029803333,

%U -1328763571296,6823754590093,-35073821767334,180407337377834,-928487386730281,4780794440512601,-24625601552074341,126883328914736618

%N a(n) = (2*sqrt(7)*sin(Pi/7))^n + (-2*sqrt(7)*sin(2*Pi/7))^n + (-2*sqrt(7)*sin(4*Pi/7))^n.

%C 2*sqrt(7)*sin(Pi/7), -2*sqrt(7)*sin(2*Pi/7) and -2*sqrt(7)*sin(4*Pi/7) are roots of polynomial x^3 + 7*x^2 - 49.

%H Colin Barker, <a href="/A275830/b275830.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-7,0,49).

%F G.f.: (3 + 14*x)/(1 + 7*x - 49*x^3). - _Bruno Berselli_, Aug 11 2016

%F a(n) = -7*a(n-1) + 49*a(n-3) with n>2, a(0)=3, a(1)=-7, a(2)=49.

%F a(2*n-1) = 7^n*A215493(n). - _Kai Wang_, May 25 2017

%t RecurrenceTable[{a[0] == 3, a[1] == -7, a[2] == 49, a[n] == -7 a[n - 1] + 49 a[n - 3]}, a, {n, 0, 30}] (* _Bruno Berselli_, Aug 11 2016 *)

%o (PARI) Vec((3 + 14*x)/(1 + 7*x - 49*x^3) + O(x^30)) \\ _Colin Barker_, Aug 30 2016

%Y Cf. A108716, A215794, A275195, A274975, A275831.

%K sign,easy

%O 0,1

%A _Kai Wang_, Aug 11 2016