A113250 - OEIS

%I #23 Sep 25 2023 07:27:34

%S -1,4,32,64,-256,4096,-4096,16384,131072,262144,-1048576,16777216,

%T -16777216,67108864,536870912,1073741824,-4294967296,68719476736,

%U -68719476736,274877906944,2199023255552,4398046511104,-17592186044416,281474976710656,-281474976710656

%N Expansion of g.f. -(1 - 48*x^2 - 256*x^3) / ((1 - 4*x)*(1 + 4*x)*(1 + 4*x + 16*x^2)).

%C Previous name was: Corresponds to m = 4 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 (see A113249). Initial terms factored (without regards to sign): 1, 4, (2)^5, (2)^6,(2)^8, (2)^12, (2)^12, (2)^14, (2)^17, (2)^18, (2)^20, (2)^24, (2)^24, (2)^26, (2)^29, (2)^30, (2)^32, (2)^36.

%H Colin Barker, <a href="/A113250/b113250.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 (-4,0,64,256).

%F G.f.: -(1 - 48*x^2 - 256*x^3) / ((1 - 4*x)*(1 + 4*x)*(1 + 4*x + 16*x^2)). Corrected by _Colin Barker_, May 19 2019

%t LinearRecurrence[{-4, 0, 64, 256}, {-1, 4, 32, 64}, 25] (* _Robert P. P. McKone_, Aug 25 2023 *)

%o (PARI) Vec(-(1 - 48*x^2 - 256*x^3) / ((1 - 4*x)*(1 + 4*x)*(1 + 4*x + 16*x^2)) + O(x^25)) \\ _Colin Barker_, May 19 2019

%Y Cf. A000302, A097948, A056450, A113249, A113251, A113252, A113253, A113254, A113255, A113256.

%K easy,sign

%O 0,2

%A _Creighton Dement_, Nov 18 2005

%E New name using g.f. from _Joerg Arndt_, Aug 25 2023