P3514 [POI 2011] LIZ-Lollipop

Byteasar runs a confectionery in Byteburg. Strawberry-vanilla flavoured lollipops are the favourite of local children. These lollipops are composed of multiple segments of the same length, each segment of either strawberry or vanilla flavour. The price of the lollipop is the sum of the values of its segments, where a vanilla segment costs one bythaler, while a strawberry segments costs two. Fig. 1: An exemplary lollipop of five segments, three strawberry flavoured and two vanilla, alternately. The price of this lollipop is $8$ bythalers. Currently Byteasar is left with only one lollipop, though possibly very long. As a salesman, he knows only too well that probably no one will want to buy the whole lollipop. For this reason he thinks of breaking the lollipop at the joints of the segments in order to get a shorter lollipop. Each fragment for sale, of course, must stay in one piece. Byteasar vast experience of a salesman, as well as his understanding of children psychology, tell him that his young customers will most likely want to spend all their money on a single lollipop. With this in mind, he wonders for which values of $k$ the lollipop he has can be broken down in such a way that as a result one would get, among other pieces, a lollipop worth exactly $k$ bythalers. Naturally, he is interested in the way of breaking the lollipop as well. As this task overwhelms him, he asks you for help.

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