A171064 G.f.: -x*(x-1)*(1+x)/(1-x-7*x^2-x^3+x^4).

0, 1, 1, 7, 15, 64, 175, 631, 1905, 6433, 20224, 66529, 212625, 692119, 2226799, 7217728, 23284815, 75343591, 243328225, 786800449, 2542156800, 8217744577, 26556314401, 85835882791, 277405671375, 896595420736, 2897714688751

Offset: 0

Author

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

Keywords

• nonn
• easy

Comments

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).

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 7 (5) (2011) 1255-1277
• Index entries for linear recurrences with constant coefficients, signature (1,7,1,-1).

Crossrefs

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

Programs

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

Mathematica

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 *)
LinearRecurrence[{1,7,1,-1},{0,1,1,7},30] (* Harvey P. Dale, Nov 15 2020 *)

Formula

a(n) = +a(n-1) +7*a(n-2) +a(n-3) -a(n-4).

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]