A171065 G.f. -x*(x-1)*(1+x)/(1-x-8*x^2-x^3+x^4).

0, 1, 1, 8, 17, 81, 224, 881, 2737, 9928, 32481, 113761, 380800, 1313441, 4441121, 15215688, 51677297, 176530481, 600723424, 2049428881, 6980069457, 23799693448, 81088954561, 276417142721, 941948403200, 3210574806081

Offset: 0

Author

R. J. Mathar, at the request of R. K. Guy, Sep 03 2010

Keywords

• nonn
• easy

Comments

The member k=8 of a family of sequences starting 0,1,1,k with recurrence a(n) = a(n-1)+k*a(n-2)+a(n-3)-a(n-4).

This is the case P1 = 1, P2 = -10, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Mar 31 2014

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Hugh Williams, R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory vol. 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 linear recurrences with constant coefficients, signature (1,8,1,-1).

Crossrefs

Cf. A116201 (k=1), A105309 (k=2), A152090 (k=3), A007598 (k=4), A005178 (k=5), A003757 (k=6). A100047.

Programs

Magma

I:=[0, 1, 1, 8]; [n le 4 select I[n] else Self(n-1) + 8*Self(n-2) + Self(n-3) - Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 19 2012

Mathematica

CoefficientList[Series[-x*(x - 1)*(1 + x)/(1 - x - 8*x^2 - x^3 + x^4), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 19 2012 *)
LinearRecurrence[{1,8,1,-1},{0,1,1,8},30] (* Harvey P. Dale, Dec 27 2017 *)

Formula

a(n)= +a(n-1) +8*a(n-2) +a(n-3) -a(n-4).

From Peter Bala, Mar 31 2014: (Start)

a(n) = ( T(n,alpha) - T(n,beta) )/(alpha - beta), where alpha = (1 + sqrt(41))/4 and beta = (1 - sqrt(41))/4 and T(n,x) denotes the Chebyshev polynomial of the first kind.

a(n) = the bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, 5/2; 1, 1/2].

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

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