There are 1br ($2495) and 2br ($3295) apartments available in the complex I live in the Barron Park area, see below. The first three months of rent are currently free. The location is very convenient - an easy, 15-20 minute bike ride to campus.

1br: https://sfbay.craigslist.org/pen/apa/d/palo-alto-3-months-free-rentchariming/7190049632.html
2br: https://sfbay.craigslist.org/pen/apa/d/palo-alto-2br-2-ba-beauitfulnewly/7186812073.html

Contact the property manager through Craigslist if you're interested, or directly at pcj914@gmail.com.