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

Author

Roman Witula, Aug 03 2012

Keywords

• sign
• easy

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).

Crossrefs

Cf. A214683.

Programs

Mathematica

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

Formula

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