A038391 Expansion of (x^3+2*x+1) / ((x-1)^4*(x^2+x+1)^2).

1, 4, 7, 13, 23, 33, 48, 69, 90, 118, 154, 190, 235, 290, 345, 411, 489, 567, 658, 763, 868, 988, 1124, 1260, 1413, 1584, 1755, 1945, 2155, 2365, 2596, 2849, 3102, 3378, 3678, 3978, 4303, 4654, 5005, 5383, 5789, 6195, 6630, 7095, 7560, 8056, 8584, 9112, 9673

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Old Name was: Bisection of A028289.

References

• B. N. Cyvin et al., Enumeration of conjugated hydrocarbons..., Structural Chem., 6 (1995), 85-88, equation (8).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,-1,2,-4,2,-1,2,-1).

Crossrefs

Cf. A028289.

Programs

Mathematica

CoefficientList[Series[(x^3 + 2 x + 1)/((x - 1)^4 (x^2 + x + 1)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Oct 22 2013 *)
LinearRecurrence[{2,-1,2,-4,2,-1,2,-1},{1,4,7,13,23,33,48,69},50] (* Harvey P. Dale, Sep 22 2015 *)

Formula

G.f.: (x^3+2*x+1) / ((x-1)^4*(x^2+x+1)^2). - Colin Barker, Aug 30 2013

From Wesley Ivan Hurt, May 07 2016: (Start)

a(n) = 2*a(n-1)-a(n-2)+2*a(n-3)-4*a(n-4)+2*a(n-5)-a(n-6)+2*a(n-7)-a(n-8).

a(n) = Sum_{i=1..n+1} (1+floor((n+i+1)/3)) * (1+floor((n-i+1)/3)). (End)

Extensions

More terms from Colin Barker, Aug 30 2013

Name changed by Wesley Ivan Hurt, May 07 2016