By Ronald L Graham; Donald Ervin Knuth; Oren Patashnik
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Extra info for Concrete mathematics : a foundation for computer science
8) 26 SUMS where A(n), B(n), and C(n) are the coefficients of dependence on the general parameters 01, B, and y. The repertoire method tells us to try plugging in simple functions of n for R,, hoping to find constant parameters 01, (3, and y where the solution is especially simple. Setting R, = 1 implies LX = 1, (3 = 0, y = 0; hence A(n) = 1. Setting R, = n implies a = 0, (3 = 1, y = 0; hence B ( n ) = n. Setting R, = n2 implies a = 0, (3 = -1, y = 2; hence 2C(n) - B ( n ) = n2 and we have C(n) = (n2 +n)/2.
Mathematicians have developed a “finite calculus,” analogous to the more traditional infinite calculus, by which it’s possible to approach summation in a nice, systematic fashion. Infinite calculus is based on the properties of the derivative operator D, defined by Df(x) = :rnO f(x+ h) - f(x) h ’ Finite calculus is based on the properties of the difference operator A, defined by Af(x) = f(x + 1) -f(x). As opposed to a cassette function. 42) This is the finite analog of the derivative in which we restrict ourselves to positive integer values of h.
We have Sq = x CljClk l
Concrete mathematics : a foundation for computer science by Ronald L Graham; Donald Ervin Knuth; Oren Patashnik