OFFSET
0,2
COMMENTS
LINKS
Indranil Ghosh, Table of n, a(n) for n = 0..548
Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
R. Flórez, R. A. Higuita, and A. Mukherjee, Alternating Sums in the Hosoya Polynomial Triangle, Article 14.9.5 Journal of Integer Sequences, Vol. 17 (2014).
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (66,-1).
FORMULA
a(n) = 66*a(n-1) - a(n-2), n>=1, a(0)=1, a(-1):=0.
a(n) = S(n, 66) with S(n, x) := U(n, x/2), Chebyshev's polynomials of the second kind. See A049310.
G.f.: 1/(1-66*x+x^2).
a(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n-k, k)*66^(n-2*k).
a(n) = ((33+8*sqrt(17))^(n+1) - (33-8*sqrt(17))^(n+1))/(16*sqrt(17)).
MAPLE
seq( simplify(ChebyshevU(n, 33)), n=0..20); # G. C. Greubel, Dec 22 2019
MATHEMATICA
LinearRecurrence[{66, -1}, {1, 66}, 14] (* Ray Chandler, Aug 11 2015 *)
ChebyshevU[Range[21] -1, 33] (* G. C. Greubel, Dec 22 2019 *)
PROG
(PARI) vector( 21, n, polchebyshev(n-1, 2, 33) ) \\ G. C. Greubel, Dec 22 2019
(Magma) m:=33; I:=[1, 2*m]; [n le 2 select I[n] else 2*m*Self(n-1) -Self(n-2): n in [1..20]]; // G. C. Greubel, Dec 22 2019
(Sage) [chebyshev_U(n, 33) for n in (0..20)] # G. C. Greubel, Dec 22 2019
(GAP) m:=33;; a:=[1, 2*m];; for n in [3..20] do a[n]:=2*m*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Dec 22 2019
CROSSREFS
Chebyshev sequence U(n, m): A000027 (m=1), A001353 (m=2), A001109 (m=3), A001090 (m=4), A004189 (m=5), A004191 (m=6), A007655 (m=7), A077412 (m=8), A049660 (m=9), A075843 (m=10), A077421 (m=11), A077423 (m=12), A097309 (m=13), A097311 (m=14), A097313 (m=15), A029548 (m=16), A029547 (m=17), A144128 (m=18), A078987 (m=19), this sequence (m=33).
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Aug 31 2004
STATUS
approved