A077239 Bisection (odd part) of Chebyshev sequence with Diophantine property.

7, 37, 215, 1253, 7303, 42565, 248087, 1445957, 8427655, 49119973, 286292183, 1668633125, 9725506567, 56684406277, 330380931095, 1925601180293, 11223226150663, 65413755723685, 381259308191447, 2222142093424997, 12951593252358535, 75487417420726213

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-François 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 * A362087 A046235 A297329

Adjacent sequences: A077236 A077237 A077238 * A077240 A077241 A077242

Keyword

nonn,easy

Author

Wolfdieter Lang, Nov 08 2002

Status

approved