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The Adams–Bashforth methods with ''s'' = 1, 2, 3, 4, 5 are ({{harvnb|Hairer|Nørsett|Wanner|1993|loc=§III.1}}; {{harvnb|Butcher|2003|p=103}}):
: <math> \begin{align}
* <math> y_{n+1} &= y_n + hf(t_n, y_n) \, </math>—this\qquad\text{(This is simply the Euler method;)} \\
* <math> y_{n+2} = y_{n+1} + h\left( \tfrac32 f(t_{n+1}, y_{n+1}) - \tfrac12 f(t_n, y_n)\right); </math>
* <math> y_{n+32} &= y_{n+21} + h\left( \tfracfrac{233}{12} f(t_{n+2}, y_{n+2}) - \tfrac43 f(t_{n+1}, y_{n+1}) +- \tfracfrac{51}{122}f(t_n, y_n) \right); </math>, \\
* <math> y_{n+43} &= y_{n+32} + h\left( \tfracfrac{5523}{2412} f(t_{n+32}, y_{n+32}) - \tfrac{59}{24}frac43 f(t_{n+21}, y_{n+21}) + \tfracfrac{375}{2412} f(t_{n+1}, y_{n+1}) - \tfrac38 f(t_n, y_n) \right); </math>, \\
* <math> \begin{align} y_{n+54} &= y_{n+43} + h\biglleft( &\tfracfrac{190155}{72024} f(t_{n+43}, y_{n+43}) - \tfracfrac{138759}{36024} f(t_{n+32}, y_{n+2}) + \frac{37}{24} f(t_{n+1}, y_{n+1}) - \frac{3}{8} f(t_n, y_n) \right) , \\
y_{n+5} &= y_{n+4} + h\left( \frac{1901}{720} f(t_{n+4}, y_{n+4}) - \frac{1387}{360} f(t_{n+3}, y_{n+3}) \tfracfrac{109}{30} f(t_{n+2}, y_{n+2}) - \tfracfrac{637}{360} f(t_{n+1}, y_{n+1}) + \tfracfrac{251}{720} f(t_n, y_n) \bigrright) . \end{align} </math>
\end{align} </math>
The coefficients <math>b_j</math> can be determined as follows. Use [[polynomial interpolation]] to find the polynomial ''p'' of degree <math>s-1</math> such that
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