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Hankel determinant H_n(F_3(x)) of the sequence F_3(x).
1

%I #18 Dec 03 2017 17:07:29

%S 1,-2,-4,8,16,-32,-64,128,4864,-9728,-37888,223232,446464,-1482752,

%T 5586944,-11173888,-56557568,-2490368,-4980736,472383488,-10851713024,

%U 21703426048,90592772096,-263779778560,-10023631585280,-4209210589970432,-50541367159422976

%N Hankel determinant H_n(F_3(x)) of the sequence F_3(x).

%C See Fu and Han (2016), Section 1, for precise definition.

%H Hao Fu, G.-N. Han, <a href="https://arxiv.org/abs/1601.04370">Computer assisted proof for Apwenian sequences related to Hankel determinants</a>, arXiv preprint arXiv:1601.04370 [math.NT], 2016.

%p F_3 := proc(n)

%p option remember ;

%p local v,x;

%p v := 1-x-x^2 ;

%p g := 1;

%p for p from 0 do

%p g := g*subs(x=x^(3^p),v) ;

%p if 3^p > n then

%p break;

%p end if;

%p end do:

%p coeff(g,x,n) ;

%p end proc:

%p A278995 := proc(n)

%p local H,i,j ;

%p H := Matrix(n,n) ;

%p for i from 0 to n-1 do

%p for j from 0 to n-1 do

%p H[i+1,j+1] := F_3(i+j) ;

%p end do:

%p end do:

%p LinearAlgebra[Determinant](H) ;

%p end proc:

%p seq(A278995(n),n=1..40) ;

%t F3[n_] := F3[n] = Module[{v, x}, v[x_] := 1 - x - x^2; g = 1; For[p = 0, True, p++, g = g*v[x^(3^p)]; If[3^p>n, Break[]]]; Coefficient[g, x, n]];

%t a[n_] := Module[{H}, Do[H[i+1, j+1] = F3[i+j], {i, 0, n-1}, {j, 0, n-1}]; Det[Array[H, {n, n}]]];

%t Array[a, 40] (* _Jean-François Alcover_, Dec 03 2017, translated from Maple *)

%K sign

%O 1,2

%A _N. J. A. Sloane_, Dec 07 2016