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A171065 G.f. -x*(x-1)*(1+x)/(1-x-8*x^2-x^3+x^4). 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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).

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)

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 *)

PROG

(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

CROSSREFS

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

Sequence in context: A244792 A187987 A228684 * A134790 A177129 A177178

Adjacent sequences:  A171062 A171063 A171064 * A171066 A171067 A171068

KEYWORD

nonn,easy

AUTHOR

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

STATUS

approved

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Last modified September 21 12:13 EDT 2020. Contains 337271 sequences. (Running on oeis4.)