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a(n) = -7^n*A(2*n+1), where A(n) = A(n-1) + A(n-2) + A(n-3)/7, with A(0)=3, A(1)=1, A(2)=3.
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%I #33 Sep 08 2022 08:46:03

%S -1,-31,-609,-11711,-224833,-4315871,-82846113,-1590286719,

%T -30526618241,-585978870687,-11248256653025,-215917815567167,

%U -4144686996149441,-79560041170858591,-1527208244431770145,-29315784501060168447,-562736106255347592449

%N a(n) = -7^n*A(2*n+1), where A(n) = A(n-1) + A(n-2) + A(n-3)/7, with A(0)=3, A(1)=1, A(2)=3.

%C The Berndt-type sequence number 12 for the argument 2Pi/7

%C defined by the relation sqrt(7)*a(n) = t(1)^(2*n+1) + t(2)^(2*n+1) + t(4)^(2*n+1) = (-sqrt(7) + 4*s(1))^(2*n+1) + (-sqrt(7) + 4*s(2))^(2*n+1) + (-sqrt(7) + 4*s(4))^(2*n+1), where t(j) := tan(2*Pi*j/7) and s(j) := sin(2*Pi*j/7) (the respective sum with even powers in A108716 are given, see also A215828). We note that sqrt(7)*a(n) = B(2*n+1), where B(n) is defined in the comments to A215575. From Witula-Slota's (Section 6) and Witula's (Remark 11) papers it follows that B(n) is equal to the product (-sqrt(7))^n by the value of big omega function with index n for the argument 2*i/sqrt(7). The last value is equal to A(n). The respective recurrence relation for A(n) from the following decomposition follow (see Witula-Slota's paper for details): (X-1-2*i*d*s(1))*(X-1-2*i*d*s(2))*(X-1- 2*i*d*s(4)) = X^3 - (3+i*sqrt(7))*X^2 + (3+i*2*sqrt(7)*d)*X - (1+i*sqrt(7)*d + i*sqrt(7)*d^3), since the big omega function with index n for the argument d is equal to the sum: (1 + 2*i*d*s(1))^n + (1 + 2*i*d*s(2))^n + (1 + 2*i*d*s(4))^n and it is equal to 3 for n=0, 3 + i*sqrt(7)*d for n=1, and at last 3 + 2*i*sqrt(7)*d - 7*d^2 for n=2.

%C The sequence a(n+1)/a(n) is decreasing and convergent to (t(2))^2 = 19,195669... Moreover we have floor(a(n+1)/a(n)) = 19 for every n=1,2,...

%H Vincenzo Librandi, <a href="/A215794/b215794.txt">Table of n, a(n) for n = 0..200</a>

%H Roman Witula and Damian Slota, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Slota/witula13.html">New Ramanujan-Type Formulas and Quasi-Fibonacci Numbers of Order 7</a>, Journal of Integer Sequences, Vol. 10 (2007), Article 07.5.6

%H Roman Witula, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Witula/witula17.html">Ramanujan Type Trigonometric Formulas: The General Form for the Argument 2*Pi/7</a>, Journal of Integer Sequences, Vol. 12 (2009), Article 09.8.5

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (21,-35,7).

%F G.f.: -(1+10*x-7*x^2)/(1-21*x+35*x^2-7*x^3). [_Bruno Berselli_, Aug 30 2012]

%F a(n) = -A275195(2*n-1)/(7^n). - _Kai Wang_, Aug 02 2016

%e We have -31*sqrt(7) = t(1)^3 + t(2)^3 + t(4)^3.

%t LinearRecurrence[{21, -35, 7}, {-1, -31, -609}, 17] (* _Bruno Berselli_, Aug 30 2012 *)

%o (Magma) m:=17; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(-(1+10*x-7*x^2)/(1-21*x+35*x^2-7*x^3))); // _Bruno Berselli_, Aug 30 2012

%o (Magma) I:=[-1, -31, -609]; [n le 3 select I[n] else 21*Self(n-1)-35*Self(n-2)+7*Self(n-3): n in [1..20]]; // _Vincenzo Librandi_, Mar 19 2013

%Y Cf. A215575, A108716, A215828, A275195.

%K sign,easy

%O 0,2

%A _Roman Witula_, Aug 23 2012