A215919 a(n) = -3*a(n-1) + a(n-3), with a(0)=0, a(1)=-3, a(2)=12.

0, -3, 12, -36, 105, -303, 873, -2514, 7239, -20844, 60018, -172815, 497601, -1432785, 4125540, -11879019, 34204272, -98487276, 283582809, -816544155, 2351145189, -6769852758, 19493014119, -56127897168, 161613838746, -465348502119, 1339917609189, -3858138988821

Offset: 0

Author

Roman Witula, Aug 27 2012

Keywords

• sign
• easy

Comments

The Berndt-type sequence number 10 for the argument 2Pi/9 defined by the first trigonometric relation from the section "Formula" below. The sequence a(n) is connected with sequences A215917 and A215885 - see the respective formula.

We have A035045(n)=abs(a(n+1)/3) for every n=0,1,...,5 and A035045(7) + a(7)/3 = 1, A035045(8) - a(8)/3 = 10, A035045(9) + a(9)/3 = 63, and A035045(10) - a(10)/3 = 320 - all these four results-numbers are in A069269.

Examples

			We have a(2)=-4*a(1), a(3)=-3*a(2), a(6)/a(3) = -24.25, and a(9) = 579*a(3).
		

References

• D. Chmiela and R. Witula, Two parametric quasi-Fibonacci numbers of the ninth order, (submitted, 2012).
• R. Witula, Ramanujan type formulas for arguments 2Pi/7 and 2Pi/9, Demonstratio Math. (in press, 2012).

Links

• Index entries for linear recurrences with constant coefficients, signature (-3,0,1).

Crossrefs

Cf. A215917, A215885, A215664.

Programs

Mathematica

LinearRecurrence[{-3, 0, 1}, {0, -3, 12}, 50]

Formula

a(n) = c(1)*(-c(2))^(-n) + c(2)*(-c(4))^(-n) + c(4)*(-c(1))^(-n), where c(j) := 2*cos(2*Pi*j/9).

a(n) = A215917(n+1) + A215917(n) - 2*(-1)^n*A215885(n).

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