A098306 Unsigned member r=-6 of the family of Chebyshev sequences S_r(n) defined in A092184.

0, 1, 6, 49, 384, 3025, 23814, 187489, 1476096, 11621281, 91494150, 720331921, 5671161216, 44648957809, 351520501254, 2767515052225, 21788599916544, 171541284280129, 1350541674324486, 10632792110315761, 83711795208201600

Offset: 0

Author

Wolfdieter Lang, Oct 18 2004

Keywords

• nonn
• easy

Comments

((-1)^(n+1))*a(n) = S_{-6}(n), n>=0, defined in A092184.

This sequence is a divisibility sequence, i.e., a(n) divides a(m) whenever n divides m. Case P1 = 6, P2 = -16, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Mar 25 2014

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Peter Bala, Linear divisibility sequences and Chebyshev polynomials
• H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
• H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume
• Index entries for sequences related to Chebyshev polynomials
• Index entries for linear recurrences with constant coefficients, signature (7,7,-1).

Crossrefs

Cf. A001090, A001091, A070997, A092184, A100047.

Programs

Mathematica

a[n_] := 1/10*((4 + Sqrt[15])^n + (4 - Sqrt[15])^n - 2*(-1)^n) // Simplify; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Apr 28 2017 *)
LinearRecurrence[{7, 7, -1}, {0, 1, 6, 49, 384, 3025}, 50] (* G. C. Greubel, Aug 08 2017 *)

PARI

x='x+O('x^50); Vec(x*(1-x)/((1+x)*(1-8*x+x^2))) \\ G. C. Greubel, Aug 08 2017

Formula

a(n) = (T(n, 4)-(-1)^n)/5, with Chebyshev's polynomials of the first kind evaluated at x=4: T(n, 4)=A001091(n)=((4+sqrt(15))^n + (4-sqrt(15))^n)/2.

a(n) = 8*a(n-1) - a(n-2) + 2*(-1)^(n+1), n>=2, a(0)=0, a(1)=1.

a(n) = 7*a(n-1) + 7*a(n-2) - a(n-3), n>=3, a(0)=0, a(1)=1, a(2)=6.

G.f.: x*(1-x)/((1+x)*(1-8*x+x^2)) = x*(1-x)/(1-7*x-7*x^2+x^3) (from the Stephan link, see A092184).

From Peter Bala, Mar 25 2014: (Start)

a(2*n) = 6*A001090(n)^2; a(2*n+1) = A070997(n)^2.

a(n) = |u(n)|^2, where {u(n)} is the Lucas sequence in the quadratic integer ring Z[sqrt(-6)] defined by the recurrence u(0) = 0, u(1) = 1, u(n) = sqrt(-6)*u(n-1) - u(n-2) for n >= 2.

Equivalently, a(n) = U(n-1,sqrt(-6)/2)*U(n-1,-sqrt(-6)/2), where U(n,x) denotes the Chebyshev polynomial of the second kind.

a(n) = 1/10*( (4 + sqrt(15))^n + (4 - sqrt(15))^n - 2*(-1)^n ).

a(n) = the bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, 4; 1, 3] and T(n,x) denotes the Chebyshev polynomial of the first kind.

See the remarks in A100047 for the general connection between Chebyshev polynomials of the first kind and 4th-order linear divisibility sequences. (End)