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Each row shows the state evolving until it repeats. The top row shows a generator with m = 9, a = 2, c = 0, and a seed of 1, which produces a cycle of length 6. The second row is the same generator with a seed of 3, which produces a cycle of length 2. Using a = 4 and c = 1 (bottom row) gives a cycle length of 9 with any seed in [0, 8].
This power of 2 is multiplied (arithmetic modulo 2 32) by the de Bruijn sequence, thus producing a 32-bit product in which the bit sequence of the 5 MSBs is unique for each power of 2. The 5 MSBs are shifted into the LSB positions to produce a hash code in the range [0, 31], which is then used as an index into hash table BitPositionLookup.
In mathematics and computer science, Recamán's sequence [1] [2] is a well known sequence defined by a recurrence relation. Because its elements are related to the previous elements in a straightforward way, they are often defined using recursion.
In binary (base-2) math, multiplication by a power of 2 is merely a register shift operation. Thus, multiplying by 2 is calculated in base-2 by an arithmetic shift. The factor (2 −1) is a right arithmetic shift, a (0) results in no operation (since 2 0 = 1 is the multiplicative identity element), and a (2 1) results in a left arithmetic shift ...
This method is an efficient variant of the 2 k-ary method. For example, to calculate the exponent 398, which has binary expansion (110 001 110) 2, we take a window of length 3 using the 2 k-ary method algorithm and calculate 1, x 3, x 6, x 12, x 24, x 48, x 49, x 98, x 99, x 198, x 199, x 398.
Karatsuba's basic step works for any base B and any m, but the recursive algorithm is most efficient when m is equal to n/2, rounded up. In particular, if n is 2 k , for some integer k , and the recursion stops only when n is 1, then the number of single-digit multiplications is 3 k , which is n c where c = log 2 3.
The sum of the reciprocals of the squared powers of two (powers of four) is 1/3. The smallest natural power of two whose decimal representation begins with 7 is [11] = Every power of 2 (excluding 1) can be written as the sum of four square numbers in 24 ways. The powers of 2 are the natural numbers greater than 1 that can be written as the sum ...
Levinson recursion or Levinson–Durbin recursion is a procedure in linear algebra to recursively calculate the solution to an equation involving a Toeplitz matrix. The algorithm runs in Θ ( n 2 ) time, which is a strong improvement over Gauss–Jordan elimination , which runs in Θ( n 3 ).