A077236 a(n) = 4*a(n-1) - a(n-2) with a(0) = 4 and a(1) = 11.

4, 11, 40, 149, 556, 2075, 7744, 28901, 107860, 402539, 1502296, 5606645, 20924284, 78090491, 291437680, 1087660229, 4059203236, 15149152715, 56537407624, 211000477781, 787464503500, 2938857536219, 10967965641376

Offset: 0

Author

Wolfdieter Lang, Nov 08 2002

Keywords

• nonn
• easy

Comments

a(n)^2 - 3*b(n)^2 = 13, with the companion sequence b(n)= A054491(n).

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

The odd part is A077235(n) with Diophantine companion A077234(n).

Examples

			11 = a(1) = sqrt(3*A054491(1)^2 + 13) = sqrt(3*6^2 + 13)= sqrt(121) = 11.
		

Links

• Luigi Cerlienco, Maurice Mignotte, and F. Piras, Suites récurrentes linéaires: Propriétés algébriques et arithmétiques, L'Enseignement Math., 33 (1987), 67-108. See Example 2, page 93.
• 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. A077238 (even and odd parts), A077235, A053120.

Programs

GAP

a:=[4,11];; for n in [3..30] do a[n]:=4*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Apr 28 2019

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (4-5*x)/(1-4*x+x^2) )); // G. C. Greubel, Apr 28 2019

Mathematica

CoefficientList[Series[(4-5*x)/(1-4*x+x^2), {x,0,20}], x] (* or *) LinearRecurrence[{4,-1}, {4,11}, 30] (* G. C. Greubel, Apr 28 2019 *)

PARI

my(x='x+O('x^30)); Vec((4-5*x)/(1-4*x+x^2)) \\ G. C. Greubel, Apr 28 2019

PARI

a(n) = polchebyshev(n+1, 1, 2) + 2*polchebyshev(n, 1, 2); \\ Michel Marcus, Oct 13 2021

Sage

((4-5*x)/(1-4*x+x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Apr 28 2019

Formula

a(n) = T(n+1,2) + 2*T(n,2), with T(n,x) Chebyshev's polynomials of the first kind, A053120. T(n,2) = A001075(n).

G.f.: (4-5*x)/(1-4*x+x^2).

From Al Hakanson (hawkuu(AT)gmail.com), Jul 06 2009: (Start)

a(n) = ((4+sqrt(3))*(2+sqrt(3))^n + (4-sqrt(3))*(2-sqrt(3))^n)/2. Offset 0.

a(n) = second binomial transform of 4,3,12,9,36. (End)

a(n) = (A054491(n+1) - A054491(n-1))/2 = sqrt(3*A054491(n-1)*A054491(n+1) + 52), n >= 1. - Klaus Purath, Oct 12 2021

Extensions

Edited by N. J. A. Sloane, Sep 07 2018, replacing old definition with simple formula from Philippe Deléham, Nov 16 2008