A077424 Chebyshev sequence T(n,12) with Diophantine property.

1, 12, 287, 6876, 164737, 3946812, 94558751, 2265463212, 54276558337, 1300371936876, 31154649926687, 746411226303612, 17882714781360001, 428438743526336412, 10264647129850713887, 245923092372890796876

Offset: 0

Author

Wolfdieter Lang, Nov 29 2002

Keywords

• nonn
• easy

Comments

a(143+286k)-1 and a(143+286k)+1 are consecutive odd powerful numbers. See A076445. - T. D. Noe, May 04 2006

Except for the first term, positive values of x (or y) satisfying x^2 - 24xy + y^2 + 143 = 0. - Colin Barker, Feb 19 2014

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Tanya Khovanova, Recursive Sequences
• Index entries for sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (24,-1).

Crossrefs

Cf. A090732.

Programs

Magma

I:=[1,12]; [n le 2 select I[n] else 24*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Feb 21 2014

Mathematica

CoefficientList[Series[(1 - 12 x)/(1 - 24 x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 21 2014 *)
LinearRecurrence[{24,-1},{1,12},20] (* Harvey P. Dale, Jun 15 2024 *)

PARI

Vec((1-12*x)/(1-24*x+x^2) + O(x^100)) \\ Colin Barker, Feb 19 2014

Sage

[lucas_number2(n,24,1)/2 for n in range(20)] # Zerinvary Lajos, Jun 26 2008

Formula

a(n+1)^2 - 143*b(n)^2 = 1 for n>=0, with the companion sequence b(n)=A077423(n).

a(n) = 24*a(n-1) - a(n-2) for n>0, a(-1) := 12, a(0)=1.

a(n) = T(n, 12)= (S(n, 24)-S(n-2, 24))/2 = S(n, 24)-11*S(n-1, 24) with T(n, x), resp. S(n, x), Chebyshev's polynomials of the first, resp. second, kind. See A053120 and A049310. S(n, 24)=A077423(n).

a(n) = (ap^n + am^n)/2, with ap := 12+sqrt(143) and am := 12-sqrt(143).

a(n) = sum( ((-1)^k)*(n/(2*(n-k)))*binomial(n-k, k)*(2*12)^(n-2*k), k=0..floor(n/2) ) for n>=1.

a(n+1) = sqrt(1 + 143*A077423(n)^2) for n>=0.

G.f.: (1-12*x)/(1-24*x+x^2).