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By using the recurrent relation (1.34) and the initial conditions (1.35), we can construct Table 1.1 for the numerical values of the efficiency function F(n, k) of the optimal (n, k, 1)-algorithm.

Table 1.1. Arithmetic square.

By comparing Table 1.1 with the table for binomial coefficients, known as arithmetic square or Tartaglia rectangle [132], we come to the unexpected conclusion that the efficiency function of F(n, k) is expressed by using the formula for the binomial coefficients:

Niccolo Fontana Tartaglia (1499/1500–1557) (Fig. 1.10) was the famous Italian mathematician and engineer (designing fortifications) [132].

Fig. 1.10. Italian mathematician and engineer Niccoló Fontana Tartaglia.

Tartaglia published many books, including the first Italian translations of Archimedes and Euclid. He was the first one, who used mathematics for the investigation of the paths of cannonballs, known as ballistics, in his book A New Science.