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

Author

Wolfdieter Lang, Nov 08 2002

Keywords

• nonn
• easy

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).

Examples

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

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).

Programs

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

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 *)

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

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).