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

0,1

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).

Links

Table of n, a(n) for n=0..22.

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).

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

Example

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

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 *)

Prog

(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
(Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (4-5*x)/(1-4*x+x^2) )); // G. C. Greubel, Apr 28 2019
(Sage) ((4-5*x)/(1-4*x+x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Apr 28 2019
(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

Crossrefs

Cf. A077238 (even and odd parts), A077235, A053120.

Sequence in context: A149267 A149268 A214142 * A327025 A228190 A289283

Adjacent sequences: A077233 A077234 A077235 * A077237 A077238 A077239

Keyword

nonn,easy

Author

Wolfdieter Lang, Nov 08 2002

Extensions

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

Status

approved