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Partial sums of Chebyshev sequence S(n,17)= U(n,17/2)=A078366(n).
1

%I #14 Nov 20 2022 14:43:23

%S 1,18,306,5185,87840,1488096,25209793,427078386,7235122770,

%T 122570008705,2076455025216,35177165419968,595935357114241,

%U 10095723905522130,171031371036761970,2897437583719431361,49085407552193571168

%N Partial sums of Chebyshev sequence S(n,17)= U(n,17/2)=A078366(n).

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (18,-18,1).

%F a(n) = sum(S(k, 17), k=0..n) with S(k, 17) = U(k, 17/2) = A078366(k) Chebyshev's polynomials of the second kind.

%F G.f.: 1/((1-x)*(1-17*x+x^2)) = 1/(1-18*x+18*x^2-x^3).

%F a(n) = 18*a(n-1)-18*a(n-2)+a(n-3), n>=2, a(-1)=0, a(0)=1, a(1)=18.

%F a(n) = 17*a(n-1)-a(n-2)+1, n>=1, a(-1)=0, a(0)=1.

%F a(n) = (S(n+1, 17) - S(n, 17) -1)/15.

%t LinearRecurrence[{18,-18,1},{1,18,306},20] (* _Harvey P. Dale_, Nov 20 2022 *)

%Y Cf. A212336 for more sequences with g.f. of the type 1/(1-k*x+k*x^2-x^3).

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Aug 31 2004