A113255 - OEIS

%I #13 Jun 10 2024 16:39:43

%S -1,4,227,5329,-26581,206116,2391479,16785409,-174757993,2826198244,

%T 9824173259,14210785681,-287742103741,22876687229764,-22446053606113,

%U 89792737665409,5164999769137199,122161424469552196,-606821408584323661,4689875711360495569

%N Corresponds to m = 9 in a family of 4th-order linear recurrence sequences given by a(m,n) = m^4*a(n-4) + (2*m)^2*a(n-3) - 4*a(m-1), a(m,0) = -1, a(m,1) = 4, a(m,2) = -13 + 6*(m-1) + 3*(m-1)^2, a(m,3) = (-8+m^2)^2.

%C Conjecture: a(m, 2*n+1) is a perfect square for all m,n (see A113249).

%H Colin Barker, <a href="/A113255/b113255.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (-4,0,324,6561).

%F G.f.: (-1+243*x^2+6561*x^3) / ((9*x+1)*(1-9*x)*(81*x^2+4*x+1)).

%F a(n) = -4*a(n-1) + 324*a(n-3) + 6561*a(n-4) for n > 3. - _Colin Barker_, May 20 2019

%t LinearRecurrence[{-4, 0, 324, 6561}, {-1, 4, 227, 5329}, 25] (* _Paolo Xausa_, Jun 10 2024 *)

%o (PARI) Vec(-(1 - 243*x^2 - 6561*x^3) / ((1 - 9*x)*(1 + 9*x)*(1 + 4*x + 81*x^2)) + O(x^20)) \\ _Colin Barker_, May 20 2019

%Y Cf. A000302, A097948, A056450, A113249, A113250, A113251, A113252, A113253, A113254, A113256.

%K easy,sign

%O 0,2

%A _Creighton Dement_, Nov 18 2005