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%I #21 Sep 08 2022 08:45:49
%S 0,1,1,7,15,64,175,631,1905,6433,20224,66529,212625,692119,2226799,
%T 7217728,23284815,75343591,243328225,786800449,2542156800,8217744577,
%U 26556314401,85835882791,277405671375,896595420736,2897714688751
%N G.f.: -x*(x-1)*(1+x)/(1-x-7*x^2-x^3+x^4).
%C The member k=7 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).
%H Vincenzo Librandi, <a href="/A171064/b171064.txt">Table of n, a(n) for n = 0..1000</a>
%H Hugh Williams, R. K. Guy, <a href="http://dx.doi.org/10.1142/S1793042111004587">Some fourth-order linear divisibility sequences</a>, Intl. J. Number Theory 7 (5) (2011) 1255-1277
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,7,1,-1).
%F a(n) = +a(n-1) +7*a(n-2) +a(n-3) -a(n-4).
%F The roots (r1..r4) of the characteristic polynomials for this "family" of sequences have the following form (not simplified) for k= 1,2,3,4,5,6.... r1=(sqrt(4*k+10+2*sqrt(4*k+9))+sqrt(4*k-6+2*sqrt(4*k+9)))/4. r2=(sqrt(4*k+10+2*sqrt(4*k+9))-sqrt(4*k-6+2*sqrt(4*k+9)))/4. r3=(-sqrt(4*k+10-2*sqrt(4*k+9))-sqrt(4*k-6-2*sqrt(4*k+9)))/4. r4=(-sqrt(4*k+10-2*sqrt(4*k+9))+sqrt(4*k-6-2*sqrt(4*k+9)))/4. For k=1,2,3, r3 and r4 are complex . Closed-form (not simplified) is as follows for all k (note:for k1-k3 set r3 and r4 =0 and round a(n) to nearest integer): a(n)=sqrt(4*k+9)/(4*k+9)*(((r1)^n+(r2)^n)-((r3)^n+(r4)^n)). [_Tim Monahan_, Sep 17 2011]
%t CoefficientList[Series[-x*(x - 1)*(1 + x)/(1 - x - 7*x^2 - x^3 + x^4), {x, 0, 40}], x] (* _Vincenzo Librandi_, Dec 19 2012 *)
%t LinearRecurrence[{1,7,1,-1},{0,1,1,7},30] (* _Harvey P. Dale_, Nov 15 2020 *)
%o (Magma) I:=[0, 1, 1, 7]; [n le 4 select I[n] else Self(n-1) + 7*Self(n-2) + Self(n-3) - Self(n-4): n in [1..30]]; // _Vincenzo Librandi_, Dec 19 2012
%Y Cf. A116201 (k=1), A105309 (k=2), A152090 (k=3), A007598 (k=4), A005178 (k=5), A003757 (k=6).
%K nonn,easy
%O 0,4
%A _R. J. Mathar_, at the request of _R. K. Guy_, Sep 03 2010
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