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A215076 a(n) = 3*a(n-1) + 4*a(n-2) + a(n-3) with a(0)=3, a(1)=3, a(2)=17. 18
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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).

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..25.

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).

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

EXAMPLE

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.

MATHEMATICA

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

PROG

(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

CROSSREFS

Cf. A214683.

Sequence in context: A014783 A090524 A215808 * A095106 A130184 A183039

Adjacent sequences:  A215073 A215074 A215075 * A215077 A215078 A215079

KEYWORD

nonn,easy

AUTHOR

Roman Witula, Aug 02 2012

EXTENSIONS

More terms from Michel Marcus, Apr 20 2016

STATUS

approved

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Last modified November 28 11:28 EST 2020. Contains 338720 sequences. (Running on oeis4.)