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A077239 Bisection (odd part) of Chebyshev sequence with Diophantine property. 5
7, 37, 215, 1253, 7303, 42565, 248087, 1445957, 8427655, 49119973, 286292183, 1668633125, 9725506567, 56684406277, 330380931095, 1925601180293, 11223226150663, 65413755723685, 381259308191447, 2222142093424997, 12951593252358535, 75487417420726213 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

a(n)^2 - 8*b(n)^2 = 17, with the companion sequence b(n)= A077413(n).

The even part is A077240(n) with Diophantine companion A054488(n).

LINKS

Colin Barker, Table of n, a(n) for n = 0..1000

Tanya Khovanova, Recursive Sequences

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (6,-1).

FORMULA

a(n) = 6*a(n-1) - a(n-2), a(-1) := 5, a(0)=7.

a(n) = 2*T(n+1, 3)+T(n, 3), with T(n, x) Chebyshev's polynomials of the first kind, A053120. T(n, 3)= A001541(n).

G.f.: (7-5*x)/(1-6*x+x^2).

a(n) = (((3-2*sqrt(2))^n*(-8+7*sqrt(2))+(3+2*sqrt(2))^n*(8+7*sqrt(2))))/(2*sqrt(2)). - Colin Barker, Oct 12 2015

EXAMPLE

37 = a(1) = sqrt(8*A077413(1)^2 +17) = sqrt(8*13^2 + 17)= sqrt(1369) = 37.

MATHEMATICA

Table[2*ChebyshevT[n+1, 3] + ChebyshevT[n, 3], {n, 0, 19}]  (* Jean-Fran├žois Alcover, Dec 19 2013 *)

PROG

(PARI) Vec((7-5*x)/(1-6*x+x^2) + O(x^40)) \\ Colin Barker, Oct 12 2015

CROSSREFS

Cf. A077242 (even and odd parts).

Sequence in context: A126475 A274674 A255672 * A046235 A144496 A025012

Adjacent sequences:  A077236 A077237 A077238 * A077240 A077241 A077242

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Nov 08 2002

STATUS

approved

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Last modified May 25 19:28 EDT 2017. Contains 287059 sequences.