A054491 a(n) = 4*a(n-1) - a(n-2), a(0)=1, a(1)=6.

1, 6, 23, 86, 321, 1198, 4471, 16686, 62273, 232406, 867351, 3236998, 12080641, 45085566, 168261623, 627960926, 2343582081, 8746367398, 32641887511, 121821182646, 454642843073, 1696750189646, 6332357915511, 23632681472398

Offset

0,2

Comments

Bisection (even part) of Chebyshev sequence with Diophantine property.

The odd part is A077234 with Diophantine companion A077235.

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 sequences related to Chebyshev polynomials.

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

Cf. A001353, A001834, A077234, A077235.

Sequence in context: A006815 A264690 A241394 * A282710 A295132 A013261

Adjacent sequences: A054488 A054489 A054490 * A054492 A054493 A054494

Keyword

easy,nonn

Author

Barry E. Williams, May 04 2000

Extensions

Chebyshev comments from Wolfdieter Lang, Nov 08 2002

Status

approved