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A113250
Expansion of g.f. -(1 - 48*x^2 - 256*x^3) / ((1 - 4*x)*(1 + 4*x)*(1 + 4*x + 16*x^2)).
8
-1, 4, 32, 64, -256, 4096, -4096, 16384, 131072, 262144, -1048576, 16777216, -16777216, 67108864, 536870912, 1073741824, -4294967296, 68719476736, -68719476736, 274877906944, 2199023255552, 4398046511104, -17592186044416, 281474976710656, -281474976710656
OFFSET
0,2
COMMENTS
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.
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.
FORMULA
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
MATHEMATICA
LinearRecurrence[{-4, 0, 64, 256}, {-1, 4, 32, 64}, 25] (* Robert P. P. McKone, Aug 25 2023 *)
PROG
(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
KEYWORD
easy,sign
AUTHOR
Creighton Dement, Nov 18 2005
EXTENSIONS
New name using g.f. from Joerg Arndt, Aug 25 2023
STATUS
approved