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A077246 Bisection (even part) of Chebyshev sequence with Diophantine property. 4
2, 13, 102, 803, 6322, 49773, 391862, 3085123, 24289122, 191227853, 1505533702, 11853041763, 93318800402, 734697361453, 5784260091222, 45539383368323, 358530806855362, 2822707071474573, 22223125764941222 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

3*a(n)^2 - 5*b(n)^2 = 7, with the companion sequence b(n)= A077245(n).

The odd part is A077244(n) with Diophantine companion A077243(n).

LINKS

Table of n, a(n) for n=0..18.

Tanya Khovanova, Recursive Sequences

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (8,-1).

FORMULA

a(n)= 8*a(n-1) - a(n-2), a(-1) := 3, a(0)=2.

a(n)= (T(n+1, 4)+2*T(n, 4))/3, with T(n, x) Chebyshev's polynomials of the first kind, A053120. T(n, 4)= A001091(n).

G.f.: (2-3*x)/(1-8*x+x^2).

a(n)=[4-sqrt(15)]^n-(1/6)*[4-sqrt(15)]^n*sqrt(15)+[4+sqrt(15)]^n+(1/6)*sqrt(15)*[4 +sqrt(15)]^n, with n>=0 - Paolo P. Lava, Jul 08 2008

EXAMPLE

13 = a(1) = sqrt((5*A077245(1)^2 + 7)/3) = sqrt((5*10^2 + 7)/3) = sqrt(169) = 13.

MATHEMATICA

LinearRecurrence[{8, -1}, {2, 13}, 30] (* Harvey P. Dale, Apr 30 2012 *)

CROSSREFS

Sequence in context: A030519 A141116 A234299 * A266906 A107000 A046891

Adjacent sequences:  A077243 A077244 A077245 * A077247 A077248 A077249

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Nov 08 2002

STATUS

approved

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Last modified November 21 22:32 EST 2017. Contains 295054 sequences.