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

0,3

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).

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)

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 *)

Prog

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

Crossrefs

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

Sequence in context: A283226 A292124 A104170 * A055847 A371406 A365193

Adjacent sequences: A098303 A098304 A098305 * A098307 A098308 A098309

Keyword

nonn,easy

Author

Wolfdieter Lang, Oct 18 2004

Status

approved