ON THE BINARY SEQUENCE $(1,1,0,1,0^3,1,0^7,1,0^{15},\ldots )$

Abstract

Abstract Let $\mathbb {F}$ be a field and $(s_0,\ldots ,s_{n-1})$ be a finite sequence of elements of $\mathbb {F}$ . In an earlier paper [G. H. Norton, ‘On the annihilator ideal of an inverse form’, J. Appl. Algebra Engrg. Comm. Comput.28 (2017), 31–78], we used the $\mathbb {F}[x,z]$ submodule $\mathbb {F}[x^{-1},z^{-1}]$ of Macaulay’s inverse system $\mathbb {F}[[x^{-1},z^{-1}]]$ (where z is our homogenising variable) to construct generating forms for the (homogeneous) annihilator ideal of $(s_0,\ldots ,s_{n-1})$ . We also gave an $\mathcal {O}(n^2)$ algorithm to compute a special pair of generating forms of such an annihilator ideal. Here we apply this approach to the sequence r of the title. We obtain special forms generating the annihilator ideal for $(r_0,\ldots ,r_{n-1})$ without polynomial multiplication or division, so that the algorithm becomes linear. In particular, we obtain its linear complexities. We also give additional applications of this approach.