OFFSET
0,2
COMMENTS
REFERENCES
A. H. Beiler, Recreations in the Theory of Numbers, Dover, N. Y., 1964, pp. 122-125, 194-196.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
I. Adler, Three Diophantine equations - Part II, Fib. Quart., 7 (1969), pp. 181-193.
E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pp. 231-242.
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (4,-1).
FORMULA
-3*a(n)^2 + A077236(n)^2 = 13.
a(n) = ( 6*((2+sqrt(3))^n-(2-sqrt(3))^n) - ((2+sqrt(3))^(n-1)-(2-sqrt(3))^(n-1)) )/(2*sqrt(3)).
a(n) = 6*S(n-1, 4) - S(n-2, 4) = S(n, 4) + 2*S(n-1, 4), with S(n, x) := U(n, x/2) Chebyshev's polynomials of 2nd kind, A049310. S(-1, x) := 0, S(-2, x) := -1, S(n, 4)= A001353(n+1).
G.f.: (1+2*x)/(1-4*x+x^2).
a(n+1) = A001353(n+2) + 2*A001353(n+1) - Creighton Dement, Nov 28 2004. Comment from Vim Wenders, Mar 26 2008: This is easily verified using a(n) = (6*( (2+sqrt(3))^n - (2-sqrt(3))^n ) - ( (2+sqrt(3))^(n-1) - (2-sqrt(3))^(n-1) ))/(2*sqrt(3)) and A001353(n) = ( (2+sqrt(3))^n - (2-sqrt(3))^n )/(2*sqrt(3)).
a(n) = (-1)^n*Sum_{k = 0..n} A238731(n,k)*(-7)^k. - Philippe Deléham, Mar 05 2014
E.g.f.: (1/3)*exp(2*x)*(3*cosh(sqrt(3)*x) + 4*sqrt(3)*sinh(sqrt(3)*x)). - Stefano Spezia, Jan 27 2020
MAPLE
seq( simplify(ChebyshevU(n, 2) +2*ChebyshevU(n-1, 2)), n=0..30); # G. C. Greubel, Jan 15 2020
MATHEMATICA
Table[ChebyshevU[n, 2] +2*ChebyshevU[n-1, 2], {n, 0, 30}] (* G. C. Greubel, Jan 15 2020 *)
LinearRecurrence[{4, -1}, {1, 6}, 30] (* Harvey P. Dale, Sep 04 2021 *)
PROG
(PARI) a(n) = if (n==0, 1, if (n==1, 6, 4*a(n-1)-a(n-2))) \\ Michel Marcus, Jun 23 2013
(PARI) a(n) = polchebyshev(n, 2, 2) + 2*polchebyshev(n-1, 2, 2); \\ Michel Marcus, Oct 13 2021
(Magma) I:=[1, 6]; [n le 2 select I[n] else 4*Self(n-1) -Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 15 2020
(Sage) [chebyshev_U(n, 2) +2*chebyshev_U(n-1, 2) for n in (0..30)]; # G. C. Greubel, Jan 15 2020
(GAP) a:=[1, 6];; for n in [3..30] do a[n]:=4*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Jan 15 2020
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Barry E. Williams, May 04 2000
EXTENSIONS
Chebyshev comments from Wolfdieter Lang, Nov 08 2002
STATUS
approved