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

%I A215100 #30 Nov 26 2022 02:47:47
%S A215100 2,5,22,88,357,1445,5851,23690,95919,388368,1572470,6366801,25778651,
%T A215100 104375627,422608286,1711106017,6928126822,28051412820,113577851765,
%U A215100 459867333397,1861964820071,7538941645566,30524551550379,123591386053472,500411306007498,2026124013786761
%N A215100 a(n) = 3*a(n-1) + 4*a(n-2) + a(n-3) with a(0)=2, a(1)=5, a(2)=22.
%C A215100 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).
%D A215100 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.
%H A215100 G. C. Greubel, <a href="/A215100/b215100.txt">Table of n, a(n) for n = 0..1000</a>
%H A215100 Roman Witula, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Witula/witula17.html">Ramanujan Type Trigonometric Formulas: The General Form for the Argument 2*Pi/7</a>, Journal of Integer Sequences, Vol. 12 (2009), Article 09.8.5.
%H A215100 Roman Witula, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Witula/witula30.html">Full Description of Ramanujan Cubic Polynomials</a>, Journal of Integer Sequences, Vol. 13 (2010), Article 10.5.7.
%H A215100 Roman Witula, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Witula2/witula40r.html">Ramanujan Cubic Polynomials of the Second Kind</a>, Journal of Integer Sequences, Vol. 13 (2010), Article 10.7.5.
%H A215100 Roman Witula, <a href="https://doi.org/10.1515/dema-2013-0418">Ramanujan Type Trigonometric Formulae</a>, Demonstratio Math. 45 (2012) 779-796.
%H A215100 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,4,1).
%F A215100 G.f.: (2 - x - x^2)/(1 - 3*x - 4*x^2 - x^3).
%e A215100 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).
%t A215100 LinearRecurrence[{3, 4, 1}, {2, 5, 22}, 40]
%o A215100 (PARI) Vec((2-x-x^2)/(1-3*x-4*x^2-x^3) + O(x^40)) \\ _Michel Marcus_, Apr 20 2016
%o A215100 (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
%o A215100 (SageMath)
%o A215100 @CachedFunction
%o A215100 def a(n): # a = A215100
%o A215100     if (n<3): return (2,5,22)[n]
%o A215100     else: return 3*a(n-1) + 4*a(n-2) + a(n-3)
%o A215100 [a(n) for n in range(41)] # _G. C. Greubel_, Nov 25 2022
%Y A215100 Cf. A006053, A214683, A215076, A120757.
%K A215100 nonn,easy
%O A215100 0,1
%A A215100 _Roman Witula_, Aug 03 2012
%E A215100 More terms from _Michel Marcus_, Apr 20 2016