OFFSET
0,2
REFERENCES
A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 122-125, 194-196.
LINKS
Vincenzo Librandi, 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
a(n) = (7*((2+sqrt(3))^n - (2-sqrt(3))^n) - ((2+sqrt(3))^(n-1) - (2-sqrt(3))^(n-1)))/2*sqrt(3).
a(n) = 4*a(n-1) - a(n-2), a(0)=1, a(0)=7.
a(n) = ChebyshevU(n,2) + 3*Chebyshev(n-1,2) = ChebyshevT(n,2) + 5*ChebyshevU(n-1,2). - G. C. Greubel, Jan 19 2020
MAPLE
seq( simplify(ChebyshevU(n, 2) +3*ChebyshevU(n-1, 2)), n=0..30); # G. C. Greubel, Jan 19 2020
MATHEMATICA
LinearRecurrence[{4, -1}, {1, 7}, 40] (* Vincenzo Librandi, Jun 23 2012 *)
Table[ChebyshevU[n, 2] +3*ChebyshevU[n-1, 2], {n, 0, 30}] (* G. C. Greubel, Jan 19 2020 *)
PROG
(Magma) I:=[1, 7]; [n le 2 select I[n] else 4*Self(n-1)-Self(n-2): n in[1..30]]; // Vincenzo Librandi, Jun 23 2012
(PARI) Vec((1+3*x)/(1-4*x+x^2) + O(x^30)) \\ Michel Marcus, Mar 20 2015
(PARI) vector(31, n, polchebyshev(n-1, 1, 2) +5*polchebyshev(n-2, 2, 2) ) \\ G. C. Greubel, Jan 19 2020
(Sage) [chebyshev_U(n, 2) + 3*chebyshev_U(n-1, 2) for n in (0..30)] # G. C. Greubel, Jan 19 2020
(GAP) a:=[1, 7];; for n in [3..30] do a[n]:=4*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Jan 19 2020
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Barry E. Williams, May 06 2000
STATUS
approved