|
|
A097316
|
|
Chebyshev U(n,x) polynomial evaluated at x=33.
|
|
22
|
|
|
1, 66, 4355, 287364, 18961669, 1251182790, 82559102471, 5447649580296, 359462313197065, 23719065021425994, 1565098829100918539, 103272803655639197580, 6814439942443086121741, 449649763397588044837326
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
Used to form integer solutions of Pell equation a^2 - 17*b^2 =-1. See A078989 with A078988.
|
|
LINKS
|
|
|
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 *)
|
|
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
|
|
|
STATUS
|
approved
|
|
|
|