A278995 Hankel determinant H_n(F_3(x)) of the sequence F_3(x).

1, -2, -4, 8, 16, -32, -64, 128, 4864, -9728, -37888, 223232, 446464, -1482752, 5586944, -11173888, -56557568, -2490368, -4980736, 472383488, -10851713024, 21703426048, 90592772096, -263779778560, -10023631585280, -4209210589970432, -50541367159422976

Offset

1,2

Comments

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

Links

Table of n, a(n) for n=1..27.

Hao Fu, G.-N. Han, Computer assisted proof for Apwenian sequences related to Hankel determinants, arXiv preprint arXiv:1601.04370 [math.NT], 2016.

Maple

F_3 := proc(n)
    option remember ;
    local v, x;
    v := 1-x-x^2 ;
    g := 1;
    for p from 0 do
        g := g*subs(x=x^(3^p), v) ;
        if 3^p > n then
            break;
        end if;
    end do:
    coeff(g, x, n) ;
end proc:
A278995 := proc(n)
    local H, i, j ;
    H := Matrix(n, n) ;
    for i from 0 to n-1 do
    for j from 0 to n-1 do
        H[i+1, j+1] := F_3(i+j) ;
    end do:
    end do:
    LinearAlgebra[Determinant](H) ;
end proc:
seq(A278995(n), n=1..40) ;

Mathematica

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]];
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}]]];
Array[a, 40] (* Jean-François Alcover, Dec 03 2017, translated from Maple *)

Crossrefs

Sequence in context: A079838 A109912 A079845 * A117302 A265407 A023422

Adjacent sequences: A278992 A278993 A278994 * A278996 A278997 A278998

Keyword

sign

Author

N. J. A. Sloane, Dec 07 2016

Status

approved