OFFSET
0,1
COMMENTS
Lim_{n-> infinity} a(n)/a(n+1) = 0.0622577... = 1/(8+sqrt(65)) = (sqrt(65)-8).
Lim_{n-> infinity} a(n+1)/a(n) = 16.0622577... = (8+sqrt(65)) = 1/(sqrt(65)-8).
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..500
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (16,1).
FORMULA
a(n) = 16*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 16.
a(n) = (8+sqrt(65))^n + (8-sqrt(65))^n.
a(n)^2 = a(2n) - 2 if n = 1, 3, 5, ...;
a(n)^2 = a(2n) + 2 if n = 2, 4, 6, ....
G.f.: (2-16*x)/(1-16*x-x^2). - Philippe Deléham, Nov 02 2008
a(n) = Lucas(n, 16) = 2*(-i)^n * ChebyshevT(n, 8*i). - G. C. Greubel, Dec 31 2019
E.g.f.: 2*exp(8*x)*cosh(sqrt(65)*x). - Stefano Spezia, Jan 01 2020
EXAMPLE
a(4) = 16*a(3) + a(2) = 16*4144 + 258 = (8+sqrt(65))^4 + (8-sqrt(65))^4 = 66561.99998497... + 0.00001502... = 66562.
MAPLE
seq(simplify(2*(-I)^n*ChebyshevT(n, 8*I)), n = 0..20); # G. C. Greubel, Dec 31 2019
MATHEMATICA
LinearRecurrence[{16, 1}, {2, 16}, 40] (* or *) With[{c=Sqrt[65]}, Simplify/@ Table[(c-8)((8+c)^n-(8-c)^n (129+16c)), {n, 20}]] (* Harvey P. Dale, Oct 31 2011 *)
LucasL[Range[20]-1, 16] (* G. C. Greubel, Dec 31 2019 *)
PROG
(PARI) vector(21, n, 2*(-I)^(n-1)*polchebyshev(n-1, 1, 8*I) ) \\ G. C. Greubel, Dec 31 2019
(Magma) m:=16; I:=[2, m]; [n le 2 select I[n] else m*Self(n-1) +Self(n-2): n in [1..20]]; // G. C. Greubel, Dec 31 2019
(Sage) [2*(-I)^n*chebyshev_T(n, 8*I) for n in (0..20)] # G. C. Greubel, Dec 31 2019
(GAP) m:=16;; a:=[2, m];; for n in [3..20] do a[n]:=m*a[n-1]+a[n-2]; od; a; # G. C. Greubel, Dec 31 2019
CROSSREFS
Lucas polynomials: A114525.
Lucas polynomials Lucas(n,m): A000032 (m=1), A002203 (m=2), A006497 (m=3), A014448 (m=4), A087130 (m=5), A085447 (m=6), A086902 (m=7), A086594 (m=8), A087798 (m=9), A086927 (m=10), A001946 (m=11), A086928 (m=12), A088316 (m=13), A090300 (m=14), A090301 (m=15), this sequence (m=16), A090306 (m=17), A090307 (m=18), A090308 (m=19), A090309 (m=20), A090310 (m=21), A090313 (m=22), A090314 (m=23), A090316 (m=24), A330767 (m=25), A087281 (m=29), A087287 (m=76), A089772 (m=199).
KEYWORD
easy,nonn
AUTHOR
Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 25 2004
EXTENSIONS
More terms from Ray Chandler, Feb 14 2004
STATUS
approved