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a(n) = -2*a(n-1) + a(n-2) + a(n-3) with a(0)=a(1)=-1, a(2)=1.
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%I #24 Oct 03 2019 04:05:03

%S -1,-1,1,-4,8,-19,42,-95,213,-479,1076,-2418,5433,-12208,27431,-61637,

%T 138497,-311200,699260,-1571223,3530506,-7932975,17825233,-40052935,

%U 89998128,-202223958,454393109,-1021012048,2294193247,-5155005433,11583192065

%N a(n) = -2*a(n-1) + a(n-2) + a(n-3) with a(0)=a(1)=-1, a(2)=1.

%C We call this sequence the Ramanujan-type sequence number 1a for the argument 2Pi/7 because it forms the negative part of A214683 (i.e. for nonpositive indices). It is interesting that the same Ramanujan-type formula (with negative powers - see comments in A214683) is connected with a(n). Indeed, we have 7^(1/3)*a(n) = (c(1)/c(2))^(1/3)*(2c(1))^(-n) + (c(2)/c(4))^(1/3)*(2c(2))^(-n) + (c(4)/c(1))^(1/3)*(2c(4))^(-n) = (c(1)/c(2))^(1/3)*(2c(2))^(-n+1) + (c(2)/c(4))^(1/3)*(2c(4))^(-n+1) + (c(4)/c(1))^(1/3)*(2c(1))^(-n+1), where c(j) := Cos(2Pi*j/7). This relation follows from the following identity: (2*c(j))^(-n-1) = (2*c(2j)+2*c(j))*(2*c(j))^(-n) =((2*c(j))^2+2*c(j)-2)*(2*c(j))^(-n) whenever j is not divided by 7 since 8*c(j)*c(2j)*c(4j)=1.

%D R. Witula, E. Hetmaniok, D. Slota, Sums of the powers of any order roots taken from the roots of a given polynomial, Proceedings of the Fifteenth International Conference on Fibonacci Numbers and Their Applications, Eger, Hungary, 2012.

%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 Roman Witula, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Witula/witula30.html">Full Description of Ramanujan Cubic Polynomials</a>, Journal of Integer Sequences, Vol. 13 (2010), Article 10.5.7.

%H Roman Witula, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Witula2/witula40r.html">Ramanujan Cubic Polynomials of the Second Kind</a>, Journal of Integer Sequences, Vol. 13 (2010), Article 10.7.5.

%H Roman Witula, <a href="https://doi.org/10.1515/dema-2013-0418">Ramanujan Type Trigonometric Formulae</a>, Demonstratio Math. 45 (2012) 779-796.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-2,1,1).

%F G.f.: (1+3*x)/(-1-2*x+x^2+x^3).

%t LinearRecurrence[{-2, 1, 1}, {-1, -1, 1}, 40]

%Y Cf. A214683.

%K sign,easy

%O 0,4

%A _Roman Witula_, Aug 03 2012