A077240 Bisection (even part) of Chebyshev sequence with Diophantine property.

5, 23, 133, 775, 4517, 26327, 153445, 894343, 5212613, 30381335, 177075397, 1032071047, 6015350885, 35060034263, 204344854693, 1191009093895, 6941709708677, 40459249158167, 235813785240325, 1374423462283783, 8010726988462373, 46689938468490455

Offset: 0

Author

Wolfdieter Lang, Nov 08 2002

Keywords

• nonn
• easy

Comments

a(n)^2 - 8*b(n)^2 = 17, with the companion sequence b(n)= A054488(n).

The odd part is A077239(n) with Diophantine companion A077413(n).

Examples

			23 = a(1) = sqrt(8*A054488(1)^2 + 17) = sqrt(8*8^2 + 17)= sqrt(529) = 23.
		

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).

Crossrefs

Cf. A077242 (even and odd parts).

Programs

Mathematica

Table[ChebyshevT[n+1, 3] + 2*ChebyshevT[n, 3], {n, 0, 19}]  (* Jean-François Alcover, Dec 19 2013 *)
LinearRecurrence[{6,-1},{5,23},30] (* Harvey P. Dale, Mar 29 2017 *)

PARI

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

Formula

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

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

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

a(n) = (((3-2*sqrt(2))^n*(-4+5*sqrt(2))+(3+2*sqrt(2))^n*(4+5*sqrt(2))))/(2*sqrt(2)). - Colin Barker, Oct 12 2015