A055845 a(n) = 4*a(n-1) - a(n-2) with a(0)=1, a(1)=8.

1, 8, 31, 116, 433, 1616, 6031, 22508, 84001, 313496, 1169983, 4366436, 16295761, 60816608, 226970671, 847066076, 3161293633, 11798108456, 44031140191, 164326452308, 613274669041, 2288772223856

Offset: 0

Author

Barry E. Williams, May 31 2000

Keywords

• easy
• nonn

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).

Crossrefs

Cf. A054485.

Programs

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

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

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 *)

PARI

a(n) = polchebyshev(n,2,2) + 4*polchebyshev(n-1,2,2); \\ 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

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)