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%I #29 Sep 08 2022 08:45:15
%S 0,1,2,7,24,81,274,927,3136,10609,35890,121415,410744,1389537,4700770,
%T 15902591,53798080,181997601,615693474,2082876103,7046319384,
%U 23837527729,80641778674,272809183135,922906855808,3122171529233
%N Bisection of tribonacci numbers.
%C Binomial transform of A099462.
%C From _Paul Barry_, Feb 07 2006: (Start)
%C a(n+1) gives row sums of number triangle A114123 or A184883.
%C Partial sums are A113300. (End)
%H G. C. Greubel, <a href="/A099463/b099463.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,1,1).
%F G.f.: x*(1-x)/(1-3*x-x^2-x^3).
%F a(n) = Sum_{k=0..n} binomial(n, k)*Sum_{j=0..floor((k-1)/2)} binomial(j, k-2*j-1)*4^j.
%F From _Paul Barry_, Feb 07 2006: (Start)
%F a(n) = 3*a(n-1) + a(n-2) + a(n-3).
%F a(n) = Sum_{k=0..n} Sum_{j=0..n} C(2*k, n-k-j)*C(n-k, j)*2^(n-k-j). (End)
%F a(n)/a(n-1) tends to 3.38297576..., the square of the tribonacci constant A058265. - _Gary W. Adamson_, Feb 28 2006
%F If p[1]=2, p[2]=3, p[i]=4, (i>2), and if A is Hessenberg matrix of order n defined by: A[i,j] = p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n+1) = det A. - _Milan Janjic_, May 02 2010
%t LinearRecurrence[{3,1,1},{0,1,2},30] (* or *) Join[{0},Mean/@ Partition[ LinearRecurrence[ {1,1,1},{1,1,1},60],2]] (* _Harvey P. Dale_, Apr 02 2012 *)
%o (Magma) [n le 3 select (n-1) else 3*Self(n-1) +Self(n-2) +Self(n-3): n in [1..31]]; // _G. C. Greubel_, Nov 20 2021
%o (Sage)
%o def A184883(n, k): return simplify( hypergeometric([-k, 2*(k-n)], [1], 2) )
%o def A099463(n): return sum( A184883(n, k) for k in (0..n) )
%o [0]+[A099463(n-1) for n in (1..40)] # _G. C. Greubel_, Nov 20 2021
%Y Cf. A000073, A058265, A099462, A113300, A114123, A184883.
%K easy,nonn
%O 0,3
%A _Paul Barry_, Oct 16 2004
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