OFFSET
0,2
REFERENCES
R. C. Alperin, A nonlinear recurrence and its relations to Chebyshev polynomials, Fib. Q., 58:2 (2020), 140-142.
LINKS
Colin Barker, Table of n, a(n) for n = 0..963
Index entries for linear recurrences with constant coefficients, signature (12,-12,1).
FORMULA
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.
G.f.: 1/((1-x)*(1 - 11*x + x^2)) = 1/(1 - 12*x + 12*x^2 - x^3).
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.
a(n) = 11*a(n-1) - a(n-2) + 1 with n >= 1, a(-1)=0, a(0)=1.
a(n) = (S(n+1, 11) - S(n, 11) - 1)/9.
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
MATHEMATICA
LinearRecurrence[{12, -12, 1}, {1, 12, 132}, 30] (* G. C. Greubel, May 24 2019 *)
PROG
(PARI) Vec(1/((1-x)*(1-11*x+x^2)) + O(x^30)) \\ Colin Barker, Jun 15 2015
(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
(Sage) (1/((1-x)*(1 - 11*x + x^2))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 24 2019
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Aug 31 2004
STATUS
approved