OFFSET
0,2
REFERENCES
A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (4,-1).
FORMULA
a(n) = (8*((2+sqrt(3))^n - (2-sqrt(3))^n) - ((2+sqrt(3))^(n-1) - (2-sqrt(3))^(n-1)))/(2*sqrt(3)).
G.f.: (1+4*x)/(1-4*x+x^2).
a(n)^2 = 3*A144721(n)^2 - 11. - Sture Sjöstedt, Nov 30 2011
From G. C. Greubel, Jan 20 2020: (Start)
a(n) = ChebyshevU(n,2) + 4*ChebyshevU(n-1,2).
E.g.f.: exp(2*x)*( cosh(sqrt(3)*x) + 2*sqrt(3)*sinh(sqrt(3)*x) ). (End)
MAPLE
seq( simplify(ChebyshevU(n, 2) + 4*ChebyshevU(n-1, 2)), n=0..30); # G. C. Greubel, Jan 20 2020
MATHEMATICA
LinearRecurrence[{4, -1}, {1, 8}, 30] (* Sture Sjöstedt, Nov 30 2011 *)
Table[ChebyshevU[n, 2] + 4*ChebyshevU[n-1, 2], {n, 0, 30}] (* G. C. Greubel, Jan 20 2020 *)
PROG
(PARI) a(n) = polchebyshev(n, 2, 2) + 4*polchebyshev(n-1, 2, 2); \\ G. C. Greubel, Jan 20 2020
(Magma) I:=[1, 8]; [n le 2 select I[n] else 4*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 20 2020
(Sage) [chebyshev_U(n, 2) +4*chebyshev_U(n-1, 2) for n in (0..30)] # G. C. Greubel, Jan 20 2020
(GAP) a:=[1, 8];; for n in [3..30] do a[n]:=4*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Jan 20 2020
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Barry E. Williams, May 31 2000
STATUS
approved