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

0,1

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).

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) = 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

Example

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

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 *)

Prog

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

Crossrefs

Cf. A077242 (even and odd parts).

Sequence in context: A009321 A078509 A239820 * A281231 A356010 A244786

Adjacent sequences: A077237 A077238 A077239 * A077241 A077242 A077243

Keyword

nonn,easy

Author

Wolfdieter Lang, Nov 08 2002

Status

approved