A356571 a(n) = floor(f(n)), where f(n) = n^4*(15-24*n+10*n^2) + 20*n^5*(1-n)^3 / (1-2*n(1-n)).

0, 1, -16, -318, -1895, -6936, -19313, -45055, -92831, -174433, -305249, -504751, -796967, -1210969, -1781345, -2548687, -3560063, -4869505, -6538481, -8636383, -11240999, -14439001, -18326417, -23009119, -28603295, -35235937, -43045313, -52181455, -62806631, -75095833, -89237249

Offset

0,3

Comments

Sandefur shows that if the probability of winning any particular point in a tennis match is p, the fraction of the games won would be f(p).

Links

Table of n, a(n) for n=0..30.

James Sandefur, A Geometric Series from Tennis, The College Mathematics Journal, Vol. 36, No. 3 (May, 2005), pp. 224-226.

Formula

a(n) = g(n) + h(n), where g(n) = floor(n^2 * (-4*n^3 + 10*n^2 - 5*n - 5/2) + 1) and h(n) = [1 if n=0; 2 if n=1; -1 if n=3,5,6; 0 if n=4 or n>6]

Mathematica

a[n_] := Floor[n^4*(15 - 24*n + 10 n^2) + 20*n^5*(1 - n)^3/(1 - 2*n*(1 - n))]; Array[a, 30, 0] (* Amiram Eldar, Aug 12 2022 *)

Prog

(Python)
def a(n):
    return n**4 * (15-24*n+10*n**2) + 20*n**5 * (1-n)**3 // (1-2*n*(1-n))

Crossrefs

Sequence in context: A053856 A208266 A184757 * A231742 A027547 A024300

Adjacent sequences: A356568 A356569 A356570 * A356572 A356573 A356574

Keyword

sign

Author

Christoph B. Kassir, Aug 12 2022

Status

approved