A117691 Expansion of (4+3*x+3*x^2+2*x^3-x^5-x^6-x^7)/(1-x^4)^2.

4, 3, 3, 2, 8, 5, 5, 3, 12, 7, 7, 4, 16, 9, 9, 5, 20, 11, 11, 6, 24, 13, 13, 7, 28, 15, 15, 8, 32, 17, 17, 9, 36, 19, 19, 10, 40, 21, 21, 11, 44, 23, 23, 12, 48, 25, 25, 13, 52, 27, 27, 14, 56, 29, 29, 15, 60, 31, 31, 16, 64, 33, 33, 17, 68, 35, 35, 18, 72, 37, 37, 19, 76, 39, 39, 20

Offset

0,1

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 2, 0, 0, 0, -1).

Formula

G.f.: (4+3*x+3*x^2+2*x^3-x^5-x^6-x^7) / ((1-x)*(1+x)*(1+x^2))^2. - Colin Barker, Mar 15 2013

a(n) = 2*a(n-4) - a(n-8). - Colin Barker, Mar 15 2013

Mathematica

F= Table[Abs[Coefficient[ExpandAll[(x -(m+I*Sqrt[2*m+1])/(m+1))*(x -( m-I*Sqrt[2*m+1])/(m+1))], x]], {m, 100}];
Table[{Numerator[F[[n]]], Denominator[F[[n]]]}, {n, 2, 100}]//Flatten
(* Additional programs *)
t= {4, 3, 3, 2, 8, 5, 5, 3}; Do[AppendTo[t, 2*t[[-4]] - t[[-8]]], {100}]; t (* T. D. Noe, Mar 17 2013 *)
LinearRecurrence[{0, 0, 0, 2, 0, 0, 0, -1}, {4, 3, 3, 2, 8, 5, 5, 3}, 80] (* Harvey P. Dale, Jan 30 2021 *)

Prog

(Magma) I:=[4, 3, 3, 2, 8, 5, 5, 3]; [n le 8 select I[n] else 2*Self(n-4) - Self(n-8): n in [1..80]]; // G. C. Greubel, Jul 21 2023
(SageMath)
@CachedFunction
def a(n): # a = A117691
    if (n<8): return (4, 3, 3, 2, 8, 5, 5, 3)[n]
    else: return 2*a(n-4) - a(n-8)
[a(n) for n in range(81)] # G. C. Greubel, Jul 21 2023

Crossrefs

Sequence in context: A117323 A016502 A305743 * A375311 A243564 A358329

Adjacent sequences: A117688 A117689 A117690 * A117692 A117693 A117694

Keyword

nonn,easy

Author

Roger L. Bagula, Apr 12 2006

Extensions

New name from Colin Barker, Mar 17 2013

Status

approved