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a(n) = (25*n^2 - 15*n + 2)/2.
1

%I #24 Oct 25 2024 17:09:38

%S 1,6,36,91,171,276,406,561,741,946,1176,1431,1711,2016,2346,2701,3081,

%T 3486,3916,4371,4851,5356,5886,6441,7021,7626,8256,8911,9591,10296,

%U 11026,11781,12561,13366,14196,15051,15931,16836,17766,18721,19701

%N a(n) = (25*n^2 - 15*n + 2)/2.

%C The old definition of this sequence was "Generalized polygonal numbers".

%C Row T(5,n) of A080853.

%C Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]=5, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n >= 3, a(n-1)=coeff(charpoly(A,x),x^(n-2)). - _Milan Janjic_, Jan 27 2010

%H M. Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Janjic/janjic33.html">Hessenberg Matrices and Integer Sequences</a>, Journal of Integer Sequences, Vol. 13 (2010), Article 10.7.8.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F G.f.: (1+3*x+21*x^2)/(1-x)^3

%F a(n) = 25*n + a(n-1) - 20 with n > 0, a(0)=1. - _Vincenzo Librandi_, Aug 08 2010

%F From _Elmo R. Oliveira_, Oct 25 2024: (Start)

%F E.g.f.: exp(x)*(1 + 5*x + 25*x^2/2).

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

%t Table[(25n^2-15n+2)/2,{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{1,6,36},50] (* _Harvey P. Dale_, Aug 14 2018 *)

%o (PARI) a(n)=(25*n^2-15*n+2)/2 \\ _Charles R Greathouse IV_, Jun 17 2017

%K nonn,easy

%O 0,2

%A _Paul Barry_, Feb 23 2003

%E Definition replaced with the closed form by _Bruno Berselli_, Jan 16 2013