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%I #39 Sep 08 2022 08:45:14
%S 1,12,132,1441,15720,171480,1870561,20404692,222581052,2427986881,
%T 26485274640,288910034160,3151525101121,34377866078172,
%U 375005001758772,4090677153268321,44622443684192760,486756203372852040,5309695793417179681,57919897524216124452
%N Partial sums of Chebyshev sequence S(n,11) = U(n,11/2) = A004190(n).
%D R. C. Alperin, A nonlinear recurrence and its relations to Chebyshev polynomials, Fib. Q., 58:2 (2020), 140-142.
%H Colin Barker, <a href="/A097826/b097826.txt">Table of n, a(n) for n = 0..963</a>
%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 (12,-12,1).
%F a(n) = Sum_{k=0..n} S(k, 11), with S(k, 11) = U(k, 11/2) = A004190(k) Chebyshev's polynomials of the second kind.
%F G.f.: 1/((1-x)*(1 - 11*x + x^2)) = 1/(1 - 12*x + 12*x^2 - x^3).
%F a(n) = 12*a(n-1) - 12*a(n-2) + a(n-3) with n >= 2, a(-1)=0, a(0)=1, a(1)=12.
%F a(n) = 11*a(n-1) - a(n-2) + 1 with n >= 1, a(-1)=0, a(0)=1.
%F a(n) = (S(n+1, 11) - S(n, 11) - 1)/9.
%F a(n) = (2^(-n)*(-13*2^n + (65 - 18*sqrt(13))*(11 - 3*sqrt(13))^n + (11 + 3*sqrt(13))^n*(65 + 18*sqrt(13))))/117. - _Colin Barker_, Mar 06 2016
%t LinearRecurrence[{12,-12,1}, {1,12,132}, 30] (* _G. C. Greubel_, May 24 2019 *)
%o (PARI) Vec(1/((1-x)*(1-11*x+x^2)) + O(x^30)) \\ _Colin Barker_, Jun 15 2015
%o (Magma) I:=[1,12,132]; [n le 3 select I[n] else 12*Self(n-1)-12*Self(n-2) + Self(n-3): n in [1..30]]; // _G. C. Greubel_, May 24 2019
%o (Sage) (1/((1-x)*(1 - 11*x + x^2))).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, May 24 2019
%o (GAP) a:=[1,12,132];; for n in [4..30] do a[n]:=12*a[n-1]-12*a[n-2]+ a[n-3]; od; a; # _G. C. Greubel_, May 24 2019
%Y Cf. A076765, A097784.
%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
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