A097784 Partial sums of Chebyshev sequence S(n,10) = U(n,5) = A004189(n+1).
1, 11, 110, 1090, 10791, 106821, 1057420, 10467380, 103616381, 1025696431, 10153347930, 100507782870, 994924480771, 9848737024841, 97492445767640, 965075720651560, 9553264760747961, 94567571886828051, 936122454107532550, 9266656969188497450
Offset: 0
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- D. Fortin, B-spline Toeplitz Inverse Under Corner Perturbations, International Journal of Pure and Applied Mathematics, Volume 77, No. 1, 2012, 107-118. - From _N. J. A. Sloane_, Oct 22 2012
- Index entries for sequences related to Chebyshev polynomials.
- Index entries for linear recurrences with constant coefficients, signature (11,-11,1).
Programs
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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
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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
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Mathematica
LinearRecurrence[{11,-11,1}, {1,11,110}, 30] (* G. C. Greubel, May 24 2019 *) CoefficientList[Series[1/(1-11x+11x^2-x^3),{x,0,30}],x] (* Harvey P. Dale, Aug 24 2021 *) -
PARI
Vec(1/((1-x)*(1-10*x+x^2)) + O(x^30)) \\ Colin Barker, Jun 14 2015
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Sage
(1/((1-x)*(1 - 10*x + x^2))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 24 2019
Formula
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.
G.f.: 1/((1-x)*(1 - 10*x + x^2)) = 1/(1 - 11*x + 11*x^2 - x^3).
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.
a(n) = 10*a(n-1) - a(n-2) + 1 with n >= 1, a(-1)=0, a(0)=1.
a(n) = (S(n+1, 10) - S(n, 10) - 1)/8.
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