A215112 a(n) = -2*a(n-1) + a(n-2) + a(n-3) with a(0)=a(1)=-1, a(2)=1.

-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

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

Table of n, a(n) for n=0..30.

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: A301981 A083579 A335714 * A340948 A265108 A328184

Adjacent sequences: A215109 A215110 A215111 * A215113 A215114 A215115

Keyword

sign,easy

Author

Roman Witula, Aug 03 2012

Status

approved