A215100 a(n) = 3*a(n-1) + 4*a(n-2) + a(n-3) with a(0)=2, a(1)=5, a(2)=22.

2, 5, 22, 88, 357, 1445, 5851, 23690, 95919, 388368, 1572470, 6366801, 25778651, 104375627, 422608286, 1711106017, 6928126822, 28051412820, 113577851765, 459867333397, 1861964820071, 7538941645566, 30524551550379, 123591386053472, 500411306007498, 2026124013786761

Offset

0,1

Comments

Ramanujan-type sequence number 4 for the argument 2*Pi/7. We have a(n)=bs(3n+2), where the sequence bs(n) and its two conjugate sequences as(n) and cs(n) are defined in the comments to A214683 (see also A215076, A120757, A006053). Since we also have as(3n+2)=cs(3n+2)=0 from the formula for S(n) (see Comments at A214683) we obtain the relation 7^(1/3)*a(n)= (c(1)/c(4))^(n + 2/3) + (c(4)/c(2))^(n + 2/3) + (c(2)/c(1))^(n + 2/3).

References

R. Witula, E. Hetmaniok and 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).

Formula

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

Example

From 4*a(2) = a(3) = 88 we get 88*7^(1/3) = 4*((c(1)/c(4))^(8/3) + (c(4)/c(2))^(8/3) + (c(2)/c(1))^(8/3))=(c(1)/c(4))^(11/3) + (c(4)/c(2))^(11/3) + (c(2)/c(1))^(11/3).

Mathematica

LinearRecurrence[{3, 4, 1}, {2, 5, 22}, 40]

Prog

(PARI) Vec((2-x-x^2)/(1-3*x-4*x^2-x^3) + O(x^40)) \\ Michel Marcus, Apr 20 2016
(Magma) I:=[2, 5, 22]; [n le 3 select I[n] else 3*Self(n-1) +4*Self(n-2) +Self(n-3): n in [1..41]]; // G. C. Greubel, Nov 25 2022
(SageMath)
@CachedFunction
def a(n): # a = A215100
    if (n<3): return (2, 5, 22)[n]
    else: return 3*a(n-1) + 4*a(n-2) + a(n-3)
[a(n) for n in range(41)] # G. C. Greubel, Nov 25 2022

Crossrefs

Cf. A006053, A214683, A215076, A120757.

Sequence in context: A288028 A083465 A339646 * A307155 A144934 A346628

Adjacent sequences: A215097 A215098 A215099 * A215101 A215102 A215103

Keyword

nonn,easy

Author

Roman Witula, Aug 03 2012

Extensions

More terms from Michel Marcus, Apr 20 2016

Status

approved