login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Today, Nov 11 2014, is the 4th anniversary of the launch of the new OEIS web site. 70,000 sequences have been added in these four years, all edited by volunteers. Please make a donation (tax deductible in the US) to help keep the OEIS running.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A098306 Unsigned member r=-6 of the family of Chebyshev sequences S_r(n) defined in A092184. 3
0, 1, 6, 49, 384, 3025, 23814, 187489, 1476096, 11621281, 91494150, 720331921, 5671161216, 44648957809, 351520501254, 2767515052225, 21788599916544, 171541284280129, 1350541674324486, 10632792110315761, 83711795208201600 (list; graph; refs; listen; history; text; internal format)
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 4-th order linear divisibility sequences found by Williams and Guy. - Peter Bala, Mar 25 2014

LINKS

Table of n, a(n) for n=0..20.

Index entries for sequences related to Chebyshev polynomials.

_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

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 2X2 matrix T(n, M), where M is the 2X2 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 4-th order linear divisibility sequences. (End)

CROSSREFS

A001090, A070997, A100047.

Sequence in context: A027766 A097299 A104170 * A055847 A143165 A008786

Adjacent sequences:  A098303 A098304 A098305 * A098307 A098308 A098309

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Oct 18 2004

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified December 22 16:00 EST 2014. Contains 252364 sequences.