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A077236 a(n) = 4*a(n-1)-a(n-2) with a(0) = 4 and a(1) = 11. 8
4, 11, 40, 149, 556, 2075, 7744, 28901, 107860, 402539, 1502296, 5606645, 20924284, 78090491, 291437680, 1087660229, 4059203236, 15149152715, 56537407624, 211000477781, 787464503500, 2938857536219, 10967965641376 (list; graph; refs; listen; history; text; internal format)
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).

REFERENCES

Cerlienco, Luigi, Maurice Mignotte, and F. Piras. Suites récurrentes linéaires: Propriétés algébriques et aritmétiques. L'Enseignement Math., 33 (1987), 67-108. See Example 2, page 93.

LINKS

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

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).

a(n)=-(1/2)*sqrt(3)*[2-sqrt(3)]^n+(1/2)*sqrt(3)*[2+sqrt(3)]^n+2*[2-sqrt(3)]^n+2*[2 +sqrt(3)]^n, with n>=0 [From Paolo P. Lava, Nov 20 2008]

a(n)=((4+sqrt3)(2+sqrt3)^n+(4-sqrt3)(2-sqrt3)^n)/2. Offset 0. a(n)=second binomial transform of 4,3,12,9,36. [From Al Hakanson (hawkuu(AT)gmail.com), Jul 06 2009]

EXAMPLE

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

CROSSREFS

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

Sequence in context: A149267 A149268 A214142 * A228190 A289283 A152532

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

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Last modified September 22 19:04 EDT 2018. Contains 315270 sequences. (Running on oeis4.)