Jim, There is a standard method for finding an explicit formula for any sequence g(n) defined by g(n+1) = g(n)^2-g(n)+C, see my paper with Al Aho, A. V. *Aho* and N. J. A. *Sloane*, Some *doubly* exponential sequences <http://neilsloane.com/doc/doubly.html>, Fib. Quart., 11 (1973), 429-437. There is a link in https://oeis.org/A000058 (Sylvester's sequence: C=1, starting with 2). Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Sun, Jul 12, 2020 at 3:34 PM James Propp <jamespropp@gmail.com> wrote:

Kelly DeRango writes (on Twitter):

<https://twitter.com/Kderango> [image: Baseball] Kelly DeRango @Kderango <https://twitter.com/Kderango> · Jul 11 <https://twitter.com/Kderango/status/1282007791041904640> Math Twitter we need your help. Our high school math club would like to find an explicit formula for the recursive relationship g[n+1] = (g[n])² - g[n] + 5.

Can anyone help? The initial term is g[0] = 3.

The students in question believe (and may have proved) that the sum of arctan(2/g[n]) as n goes from 0 to infinity is Pi/4, so they'd like a formula for g[n].

Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun