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

Author

Barry E. Williams, May 04 2000

Keywords

• easy
• nonn

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.
• Andrej Dujella and László Szalay, Four squares from three numbers, arXiv:2506.14013 [math.NT], 2025. See p. 2.
• 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).

Crossrefs

Cf. A001353, A001834, A077234, A077235.

Programs

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

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

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 *)

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

Sage

[chebyshev_U(n,2) +2*chebyshev_U(n-1,2) for n in (0..30)]; # G. C. Greubel, Jan 15 2020

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

Extensions

Chebyshev comments from Wolfdieter Lang, Nov 08 2002