By Ronald L Graham; Donald Ervin Knuth; Oren Patashnik
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Additional info for Concrete mathematics : a foundation for computer science
One is cheap and the other is valuable. It is easy to generalize by diluting a little idea with a big terminology. It is much more dificult to prepare a refined and condensed extract from several good ingredients. - G. Pdlya 12381 f(1) = 34, f(2) = 5, f(3n) = lOf(n) + 7 6 , f(3nfl) f(3n = +2) = lOf(n)-2, lOf(n)+8, for n 3 1, for n 3 1, for n 3 1, and suppose we want to compute f (19). Here we have d = 3 and c = 10. Now 19 = (201)3, and the radix-changing solution tells us to perform a digit-bydigit replacement from radix 3 to radix 10.
There’s no problem of division by zero when p = 0, because our convention tells us that [O prime] [O < Nl/O = 0. Let’s sum up what we’ve discussed so far about sums. There are two good ways to express a sum of terms: One way uses ‘. ‘, the other uses ‘ t ‘. The three-dots form often suggests useful manipulations, particularly the combination of adjacent terms, since we might be able to spot a simplifying pattern if we let the whole sum hang out before our eyes. But too much detail can also be overwhelming.
For the time being, we’ll assume that only finitely many integers k satisfying P(k) have ok # 0; otherwise infinitely many nonzero numbers are being added together, and things can get a bit tricky. At the other extreme, if P(k) is false for all integers k, we have an “empty” sum; the value of an empty sum is defined to be zero. pCkl ak’, attaching property P(k) as a subscript of 1, so that the formula won’t stick out too much. 2) when we want to confine the notation to a single line. People are often tempted to write n-1 z k(k- l)(n- k) k=2 instead of f k(k- l)(n- k) k=O because the terms for k = 0, 1, and n in this sum are zero.
Concrete mathematics : a foundation for computer science by Ronald L Graham; Donald Ervin Knuth; Oren Patashnik