0,1

We call the sequence a(n) the Ramanujan-type sequence number 3 for the argument 2Pi/7 (see A214683 and Witula's papers for details). Since a(n)=as(3n), bs(3n)=cs(3n)=0, where the sequence as(n) and its two conjugate sequences bs(n) and cs(n) are defined in the comments to the sequence A214683 we obtain the following formula a(n) = (c(1)/c(4))^n + (c(2)/c(1))^n + (c(4)/c(2))^n, where c(j) := Cos(2*Pi*j/7). It is interesting that if we set b(n):= (c(1)/c(2))^n + (c(2)/c(4))^n + (c(4)/c(1))^n, for n=0,1,..., and we extend the definition of discussed sequence a(n) to the negative indices by the same formula, i.e.: a(n)=a(n+3)-3*a(n+2)-4*a(n+1), n=-1,-2,..., then we get b(n)=a(-n) for every n=0,1,... (see also example below).

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

G. C. Greubel, Table of n, a(n) for n = 0..1000

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 (3,4,1).

G.f.: (3-6*x-4*x^2)/(1-3*x-4*x^2-x^3).

From Kai Wang, Jul 08 2020: (Start)

a(n) = Sum_{i+2j+3k=n} 3^i*4^j*n*(i+j+k-1)!/(i!*j!*k!).

a(n) = (-1)^n*(3*A122600(n) + 6*A122600(n-1) - 4*A122600(n-2)) for n > 1. (End)

a(n) = r^n + s^n + t^n where {r,s,t} are the roots of 1+4*x+3*x^2-x^3. - Joerg Arndt, Jul 09 2020

a(n) = 3*a(n-1) + 4*a(n-2) + a(n-3). - Wesley Ivan Hurt, Jul 09 2020

We have (c(1)/c(2)) + (c(2)/c(4)) + (c(4)/c(1)) = (a(1)^2 - a(2))/2 = -4, and then (c(1)/c(2))^2 + (c(2)/c(4))^2 + (c(4)/c(1))^2 = 16 - 2*a(1) = 10.

LinearRecurrence[{3, 4, 1}, {3, 3, 17}, 40]

(PARI) Vec((-3+6*x+4*x^2)/(-1+3*x+4*x^2+x^3) + O(x^30)) \\ Michel Marcus, Apr 20 2016

(PARI) polsym(1+4*x+3*x^2-x^3, 22) \\ Joerg Arndt, Jul 09 2020

(SageMath)

@CachedFunction

def a(n): # a = A215076

if (n<3): return (3, 3, 17)[n]

else: return 3*a(n-1) + 4*a(n-2) + a(n-3)

[a(n) for n in range(40)] # G. C. Greubel, Nov 25 2022

Cf. A214683.

Sequence in context: A090524 A215808 A364736 * A095106 A130184 A183039

Adjacent sequences: A215073 A215074 A215075 * A215077 A215078 A215079

nonn,easy

Roman Witula, Aug 02 2012

More terms from Michel Marcus, Apr 20 2016

approved