Several articles deal with tilings with various colors and shapes. In this paper, we present a new type of tiling problem of a (1×n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(1\ imes n)$$\\end{document}—board where the colors have a prescribed order of preference and the size of colored dominoes is bounded by (1×s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(1\ imes s)$$\\end{document}. We show that the total number of tilings can be given as a linearly recurrent sequence of order ks, and at the same time by a higher order self-convolution of s-generalized Fibonacci sequences.