CF1045H Self-exploration

Being bored of exploring the Moon over and over again Wall-B decided to explore something he is made of — binary numbers. He took a binary number and decided to count how many times different substrings of length two appeared. He stored those values in $ c_{00} $ , $ c_{01} $ , $ c_{10} $ and $ c_{11} $ , representing how many times substrings 00, 01, 10 and 11 appear in the number respectively. For example: $ 10111100 \rightarrow c_{00} = 1, \ c_{01} = 1,\ c_{10} = 2,\ c_{11} = 3 $ $ 10000 \rightarrow c_{00} = 3,\ c_{01} = 0,\ c_{10} = 1,\ c_{11} = 0 $ $ 10101001 \rightarrow c_{00} = 1,\ c_{01} = 3,\ c_{10} = 3,\ c_{11} = 0 $ $ 1 \rightarrow c_{00} = 0,\ c_{01} = 0,\ c_{10} = 0,\ c_{11} = 0 $ Wall-B noticed that there can be multiple binary numbers satisfying the same $ c_{00} $ , $ c_{01} $ , $ c_{10} $ and $ c_{11} $ constraints. Because of that he wanted to count how many binary numbers satisfy the constraints $ c_{xy} $ given the interval $ [A, B] $ . Unfortunately, his processing power wasn't strong enough to handle large intervals he was curious about. Can you help him? Since this number can be large print it modulo $ 10^9 + 7 $ .

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Example 1: The binary numbers in the interval $ [10,1001] $ are $ 10,11,100,101,110,111,1000,1001 $ . Only number 110 satisfies the constraints: $ c_{00} = 0, c_{01} = 0, c_{10} = 1, c_{11} = 1 $ . Example 2: No number in the interval satisfies the constraints