A077409 Bisection (even part) of Chebyshev sequence with Diophantine property.

7, 59, 583, 5771, 57127, 565499, 5597863, 55413131, 548533447, 5429921339, 53750679943, 532076878091, 5267018100967, 52138104131579, 516114023214823, 5109002128016651, 50573907256951687, 500630070441500219, 4955726797158050503, 49056637901139004811

Offset

0,1

Comments

a(n)^2 - 24*b(n)^2 = 25, with the companion sequence b(n) = A077251(n).

The odd part is A077250(n) with Diophantine companion A077249(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 (10,-1).

Formula

a(n) = 10*a(n-1)- a(n-2), a(-1)=11, a(0)=7.

a(n) = T(n+1, 5)+2*T(n, 5), with T(n, x) Chebyshev's polynomials of the first kind, A053120. T(n, 5) = A001079(n).

a(n) = sqrt(24*A077251(n)^2 + 25).

G.f.: (7-11*x)/(1-10*x+x^2).

Example

59 = a(1) = sqrt(24*A077251(1)^2 + 25) = sqrt(24*12^2 + 25) = sqrt(3481) = 59.

Mathematica

CoefficientList[Series[(7 - 11 z)/(z^2 - 10 z + 1), {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 11 2011 *)
LinearRecurrence[{10, -1}, {7, 59}, 30] (* G. C. Greubel, Jan 18 2018 *)

Prog

(PARI) a(n)=if(n<0, 0, subst(poltchebi(n+1)+2*poltchebi(n), x, 5))
(PARI) Vec((7-11*x)/(1-10*x+x^2) + O(x^30)) \\ Colin Barker, Jun 15 2015
(PARI) a(n)=polchebyshev(n+1, , 5)+2*polchebyshev(n, , 5) \\ Charles R Greathouse IV, Jun 15 2015
(PARI) a(n)=([0, 1; -1, 10]^n*[7; 59])[1, 1] \\ Charles R Greathouse IV, Jun 15 2015
(Magma) I:=[7, 59]; [n le 2 select I[n] else 10*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 18 2018

Crossrefs

Sequence in context: A099659 A358599 A135150 * A192458 A203237 A099347

Adjacent sequences: A077406 A077407 A077408 * A077410 A077411 A077412

Keyword

nonn,easy

Author

Wolfdieter Lang, Nov 08, 2002

Status

approved