报 告 人: Cooper教授 (新西兰Massey大学)
时间：2011/12/30(周五)， 上午10点半-11点半
地点：闵行校区数学楼102报告厅
报告题目: Sporadic sequences and modular forms

报告内容：Just over ten years ago D. Zagier conducted an extensive computer search for integers
$a$, $b$ and $c$ such that the sequence $u_{n}$ defined by
$$
(n+1)^{2}u_{n+1}=(an^{2}+an+b)u_{n}+cn^{2}u_{n-1},\quad u_{0}=1,\;\;u_{-1}=0
$$
produces only integer values. Only six nontrivial examples were found and it is conjectured that
there are no others.

This talk is about a recent search for integral valued sequences defined by
$$
(n+1)^{3}u_{n+1}=(2n+1)(an^{2}+an+b)u_{n}+n(cn^{2}+d)u_{n-1}, \quad u_{0}=1,\;\;u_{-1}=0.
$$
Three nontrivial examples that were found are
$$
(a,b,c,d)=(13,4,27,-3),\quad (6,2,64,-4)\quad\mbox{and}\quad (14,6,-192,12).
$$
These sequences have some remarkable properties.
For example, there are explicit formulas for the terms in each sequence as sums of binomial coefficients.
There are some interesting divisibility properties that will be stated as conjectures.
Moreover, the sequences turn out to be connected with modular forms of levels $7$, $10$ and $18$, respectively.

These results, the experimental procedure that led to their discovery, and the reasons for doing the search
will be described.