宁夏回族自治区党委干部2017年第11号任前公示

百度 这样考虑的人，就是佛教徒，不是嘴上说，行动上要去做！犹如，婆罗门，月初生时。

?????????? ??????? ??????? (recurrence relation) ??????, ??? ????????????? ????? ?????? ????????? ???????? ??? ?????? ?????????? ???????? ????????, ???????????????? ??? ?????????? ??????????? ???????? ??????? ?????????????? ???????? ?????. ?????, ??????? ???????????? ??????? ??? ????????????? ?????????? ????????????????????.

??????????????????

????????? ??????

????????? ???????: ????????? ??????

????????? ??????: $1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765,10946,17711,28657,....$

?????????? ???????? ??????? ???????:

$F_{n}=F_{n-1}+F_{n-2}$

????????? ?????? ??????:

$F_{0}=0$

$F_{1}=1$

??????? ????????? ??????????:

$F_{2}=F_{1}+F_{0}=1+0=1$

$F_{3}=F_{2}+F_{1}=1+1=2$

$F_{4}=F_{3}+F_{2}=2+1=3....$

?? ????????? ?????? ??????????? ???? ??????? ??? ???????? ??????? ???????? ???????

??????????? ?????????

????????? ??????????? ??????????? ???????????? ????????? ??????????? ????????? ?????????. ??? ???????? ${\tbinom {n}{k}}$???? ???????????????. n ????????????????? k ??????????? ??????????????? ????? ???????? ??????????? ${\tbinom {n}{k}}$, ????????.

??????????? ??????????? ??????? ???????:

${\binom {n}{k}}={\binom {n-1}{k-1}}+{\binom {n-1}{k}},$

?????????? ?????? ???????: ${\tbinom {n}{0}}={\tbinom {n}{n}}=1$.

????????? ????????????? ??? ???????????? ??? ?????????? ??????? ????????? ???????? ???????????? ???????? ?????????????

?????? ??????????? $k=1,2,...$ ???? ????????? ??????????? ??????????? ???, ??? ????????? ??????????? ?????????.

${\binom {n}{k}}={\frac {n!}{k!(n-k)!}}.$ ???????????? ??????? ??????????? ??????????? ???????.

?????????

• Batchelder, Paul M. (1967). An introduction to linear difference equations. Dover Publications.
• Miller, Kenneth S. (1968). Linear difference equations. W. A. Benjamin.
• Fillmore, Jay P.; Marx, Morris L. (1968). "Linear recursive sequences". SIAM Rev. 10 (3): pp. 324-353. 
• Brousseau, Alfred (1971). Linear Recursion and Fibonacci Sequences. Fibonacci Association.
• Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 1990. ??????????? ?????????? ??? 0-262-03293-7. Chapter 4: Recurrences, pp. 62–90.
• Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren (1994). Concrete Mathematics: A Foundation for Computer Science (2 ed.). Addison-Welsey. ISBN 0-201-55802-5.
• Enders, Walter (2010). Applied Econometric Times Series (3 ed.). Archived from the original on 2025-08-06. Retrieved 2025-08-06.
• Cull, Paul; Flahive, Mary; Robson, Robbie (2005). Difference Equations: From Rabbits to Chaos. Springer. ISBN 0-387-23234-6. chapter 7.
• Jacques, Ian (2006). Mathematics for Economics and Business (Fifth ed.). Prentice Hall. pp. 551–568. ISBN 0-273-70195-9. Chapter 9.1: Difference Equations.
• Minh, Tang; Van To, Tan (2006). "Using generating functions to solve linear inhomogeneous recurrence equations" (PDF). Proc. Int. Conf. Simulation, Modelling and Optimization, SMO'06. pp. 399–404. Archived from the original (PDF) on 2025-08-06. Retrieved 2025-08-06. {{cite web}}: Cite has empty unknown parameter: |5= (help)
• Polyanin, Andrei D. "Difference and Functional Equations: Exact Solutions". at EqWorld - The World of Mathematical Equations.
• Polyanin, Andrei D. "Difference and Functional Equations: Methods". at EqWorld - The World of Mathematical Equations.

???????????????

• Hazewinkel, Michiel, ed. (2001), "Recurrence relation", Encyclopedia of Mathematics, Springer, ISBN 978-1556080104
• Weisstein, Eric W., "Recurrence Equation", MathWorld.
• Mathews, John H. "Homogeneous Difference Equations".
• Introductory Discrete Mathematics
• "OEIS Index Rec". OEIS index to a few thousand examples of linear recurrences, sorted by order (number of terms) and signature (vector of values of the constant coefficients)