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A097784 Partial sums of Chebyshev sequence S(n,10) = U(n,5) = A004189(n+1). 10

%I

%S 1,11,110,1090,10791,106821,1057420,10467380,103616381,1025696431,

%T 10153347930,100507782870,994924480771,9848737024841,97492445767640,

%U 965075720651560,9553264760747961,94567571886828051,936122454107532550,9266656969188497450

%N Partial sums of Chebyshev sequence S(n,10) = U(n,5) = A004189(n+1).

%H Colin Barker, <a href="/A097784/b097784.txt">Table of n, a(n) for n = 0..1000</a>

%H D. Fortin, <a href="http://ijpam.eu/contents/2012-77-1/11/11.pdf">B-spline Toeplitz Inverse Under Corner Perturbations</a>, International Journal of Pure and Applied Mathematics, Volume 77, No. 1, 2012, 107-118. - From _N. J. A. Sloane_, Oct 22 2012

%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 (11,-11,1).

%F a(n) = Sum_{k=0..n} S(k, 10) with S(k, 10) = U(k, 5) = A004189(k+1) Chebyshev's polynomials of the second kind.

%F G.f.: 1/((1-x)*(1 - 10*x + x^2)) = 1/(1 - 11*x + 11*x^2 - x^3).

%F a(n) = 11*a(n-1) - 11*a(n-2) + a(n-3) with n >= 2, a(-1)=0, a(0)=1, a(1)=11.

%F a(n) = 10*a(n-1) - a(n-2) + 1 with n >= 1, a(-1)=0, a(0)=1.

%F a(n) = (S(n+1, 10) - S(n, 10) - 1)/8.

%F a(n) = (-6 + (27-11*sqrt(6))*(5 - 2*sqrt(6))^n + (5 + 2*sqrt(6))^n*(27 + 11*sqrt(6)))/48. - _Colin Barker_, Mar 05 2016

%t LinearRecurrence[{11,-11,1}, {1,11,110}, 30] (* _G. C. Greubel_, May 24 2019 *)

%o (PARI) Vec(1/((1-x)*(1-10*x+x^2)) + O(x^30)) \\ _Colin Barker_, Jun 14 2015

%o (MAGMA) I:=[1,11,110]; [n le 3 select I[n] else 11*Self(n-1)-11*Self(n-2) +Self(n-3): n in [1..30]]; // _G. C. Greubel_, May 24 2019

%o (Sage) (1/((1-x)*(1 - 10*x + x^2))).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, May 24 2019

%o (GAP) a:=[1,11,110];; for n in [4..30] do a[n]:=11*a[n-1]-11*a[n-2]+ a[n-3]; od; a; # _G. C. Greubel_, May 24 2019

%Y Cf. A098296.

%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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Last modified January 28 09:17 EST 2020. Contains 331318 sequences. (Running on oeis4.)