In mathematics , the Fibonacci numbers are the numbers in the following integer sequence , called the Fibonacci sequence , and characterized by the fact that every number after the first two is the sum of the two preceding ones: {\displaystyle 1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\;\ldots } Often, especially in modern usage, the sequence is extended by one more initial term: {\displaystyle 0,\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\;\ldots } The sequence F n of Fibonacci numbers is defined by the recurrence relation :By definition, the first two numbers in the Fibonacci sequence are either 1 and 1, or 0 and 1, depending on the chosen starting point of the sequence, and each subsequent number is the sum of the previous two. {\displaystyle F_{n}=F_{n-1}+F_{n-2},} with seed values {\displaystyle F_{1}=1,\;F_{2}=1} or {\displaystyle F_{0}=0,\;F_{1}=1.} Th...