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 A215112 a(n) = -2*a(n-1) + a(n-2) + a(n-3) with a(0)=a(1)=-1, a(2)=1. 6
 -1, -1, 1, -4, 8, -19, 42, -95, 213, -479, 1076, -2418, 5433, -12208, 27431, -61637, 138497, -311200, 699260, -1571223, 3530506, -7932975, 17825233, -40052935, 89998128, -202223958, 454393109, -1021012048, 2294193247, -5155005433, 11583192065 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS 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. REFERENCES 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. LINKS Roman Witula, Ramanujan Type Trigonometric Formulas: The General Form for the Argument 2*Pi/7, Journal of Integer Sequences, Vol. 12 (2009), Article 09.8.5. Roman Witula, Full Description of Ramanujan Cubic Polynomials, Journal of Integer Sequences, Vol. 13 (2010), Article 10.5.7. Roman Witula, Ramanujan Cubic Polynomials of the Second Kind, Journal of Integer Sequences, Vol. 13 (2010), Article 10.7.5. Roman Witula, Ramanujan Type Trigonometric Formulae, Demonstratio Math. 45 (2012) 779-796. Index entries for linear recurrences with constant coefficients, signature (-2,1,1). FORMULA G.f.: (1+3*x)/(-1-2*x+x^2+x^3). MATHEMATICA LinearRecurrence[{-2, 1, 1}, {-1, -1, 1}, 40] CROSSREFS Cf. A214683. Sequence in context: A129362 A301981 A083579 * A265108 A328184 A332367 Adjacent sequences:  A215109 A215110 A215111 * A215113 A215114 A215115 KEYWORD sign,easy AUTHOR Roman Witula, Aug 03 2012 STATUS approved

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Last modified June 7 05:20 EDT 2020. Contains 334837 sequences. (Running on oeis4.)