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%I #17 Sep 08 2022 08:45:07
%S 6,0,2,12,6,20,50,56,134,264,402,836,1542,2652,5154,9392,16902,31824,
%T 58082,106172,197126,360932,662994,1223784,2245766,4130520,7606770,
%U 13976436,25711622,47310252,86978370,160002656,294324230,541249952
%N Sum of generalized tribonacci numbers (A001644) and reflected generalized tribonacci numbers (A073145).
%C Conjecture: a(n) >= 0.
%C For n > 2, a(n) is the number of cyclic sequences (q1, q2, ..., qn) consisting of zeros, ones and twos such that each triple contains 0 and 1 at least once, provided the positions of the zeros and ones are fixed on a circle. For example, a(5)=20 because only the sequences (00101), (01001), (01010), (01011), (01012), (01021), (01101), (01201), (02101), (20101) and those obtained from them by exchanging 0 and 1 contain 0 and 1 in each triple (including triples q4, q5, q1 and q5, q1, q2). For n = 1, 2 the statement is still true provided we allow the sequence to wrap around itself on a circle. E.g., a(2) = 2 since only sequences 01 and 10 can be wrapped so one obtains (010) and (101), respectively. - _Wojciech Florek_, Nov 25 2021
%H G. C. Greubel, <a href="/A075092/b075092.txt">Table of n, a(n) for n = 0..1000</a>
%H Wojciech Florek, <a href="http://doi.org/10.1016/j.amc.2018.06.014">A class of generalized Tribonacci sequences applied to counting problems</a>, Appl. Math. Comput., 338 (2018), 809-821.
%H W. O. J. Moser, <a href="http://www.fq.math.ca/Scanned/31-1/moser.pdf">Cyclic binary strings without long runs of like (alternating) bits</a>, Fibonacci Quart. 31(1) (1993), 2-6.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,4,1,0,-1).
%F a(n) = a(n-2) + 4*a(n-3) + a(n-4) - a(n-6), a(0)=6, a(1)=0, a(2)=2, a(3)=12, a(4)=6, a(5)=20.
%F G.f.: (6 - 4*x^2 - 12*x^3 - 2*x^4)/(1 - x^2 - 4*x^3 - x^4 + x^6).
%t CoefficientList[Series[(6-4*x^2-12*x^3-2*x^4)/(1-x^2-4*x^3-x^4+x^6), {x, 0, 40}], x]
%o (PARI) my(x='x+O('x^40)); Vec((6-4*x^2-12*x^3-2*x^4)/(1-x^2-4*x^3-x^4+x^6)) \\ _G. C. Greubel_, Apr 13 2019
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (6-4*x^2-12*x^3-2*x^4)/(1-x^2-4*x^3-x^4+x^6) )); // _G. C. Greubel_, Apr 13 2019
%o (Sage) ((6-4*x^2-12*x^3-2*x^4)/(1-x^2-4*x^3-x^4+x^6)).series(x, 40).coefficients(x, sparse=False) # _G. C. Greubel_, Apr 13 2019
%Y Cf. A001644, A073145, A075091, A294627.
%K easy,nonn
%O 0,1
%A Mario Catalani (mario.catalani(AT)unito.it), Aug 31 2002
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