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An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
where a represents the number of recursive calls at each level of recursion, b represents by what factor smaller the input is for the next level of recursion (i.e. the number of pieces you divide the problem into), and f(n) represents the work that the function does independently of any recursion (e.g. partitioning, recombining) at each level ...
This characterization states that a function is primitive recursive if and only if there is a natural number m such that the function can be computed by a Turing machine that always halts within A(m,n) or fewer steps, where n is the sum of the arguments of the primitive recursive function.
The function calls itself recursively on a smaller version of the input (n - 1) and multiplies the result of the recursive call by n, until reaching the base case, analogously to the mathematical definition of factorial. Recursion in computer programming is exemplified when a function is defined in terms of simpler, often smaller versions of ...
The Fibonacci sequence is constant-recursive: each element of the sequence is the sum of the previous two. Hasse diagram of some subclasses of constant-recursive sequences, ordered by inclusion In mathematics , an infinite sequence of numbers s 0 , s 1 , s 2 , s 3 , … {\displaystyle s_{0},s_{1},s_{2},s_{3},\ldots } is called constant ...
This recursive formula then allows the construction of Pascal's triangle, ... and the infinite series becomes a finite sum, thereby recovering the binomial formula.
An alternate recursive formula for the limit of ratio of two consecutive -nacci numbers can be expressed as r = ∑ k = 0 n − 1 r − k {\displaystyle r=\sum _{k=0}^{n-1}r^{-k}} . The special case n = 2 {\displaystyle n=2} is the traditional Fibonacci series yielding the golden section φ = 1 + 1 φ {\displaystyle \varphi =1+{\frac {1 ...
Te n is the sum of all products p × q where (p, q) are ordered pairs and p + q = n + 1 Te n is the number of ( n + 2)-bit numbers that contain two runs of 1's in their binary expansion. The largest tetrahedral number of the form 2 a + 3 b + 1 {\displaystyle 2^{a}+3^{b}+1} for some integers a {\displaystyle a} and b {\displaystyle b} is 8436 .