A215076 a(n) = 3*a(n-1) + 4*a(n-2) + a(n-3) with a(0)=3, a(1)=3, a(2)=17.

3, 3, 17, 66, 269, 1088, 4406, 17839, 72229, 292449, 1184102, 4794331, 19411850, 78596976, 318232659, 1288497731, 5217020805, 21123285998, 85526438945, 346289481632, 1402097486674, 5676976825495, 22985609904813, 93066834503093, 376819919954026, 1525712707779263

Offset: 0

Author

Roman Witula, Aug 02 2012

Keywords

• nonn
• easy

Comments

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).

Examples

			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.
		

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

• 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).

Crossrefs

Cf. A214683.

Programs

Mathematica

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

Formula

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

Extensions

More terms from Michel Marcus, Apr 20 2016