login
Bisection of tribonacci numbers.
9

%I #33 Oct 08 2024 18:39:14

%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 Meng-Han Wu, Henryk A. Witek, RafaƂ Podeszwa, <a href="https://match.pmf.kg.ac.rs/electronic_versions/Match93/n2/match93n2_415-462.pdf">Clar Covers and Zhang-Zhang Polynomials of Zigzag and Armchair Carbon Nanotubes</a>, MATCH Commun. Math. Comput. Chem. (2025) Vol. 93, 415-462. See p. 437.

%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