Be careful searching down the family to try to connect to your Gerrit Van Slyke. You may descend from Cornelis Antonissen or his nephew, Willem Pieterse. Cornelis came to the New World in May 1634 on board DE ENDRACHT, and settled in the Colony of Rensellaerswyck working for the Patroon. He stated in his contract that he was "about 30 years old" and a carpenter. Cornelis' son Jacques (by his Mohawk wife Ots-Toch) settled at Schenectady and had descendants who carried the Van Slyke name through his 3 sons Harmen; Cornelis; Marten.
Jacques brother Marten had no known descendants, although the man referred to as Wouter a full blooded Mohawk and Jacques' nephew in the journal kept by Jaspar Dankaarts during his travels in the Colony, may have been a son of Marten. Wouter went to the Netherlands with Jaspar and was never heard from again.
Cornelis' nephew, Willem Pieterse (aka Willem Neef, Neef meaning nephew in Dutch) settled in the Albany area when he arrived in the New World in 1660 on board DE TROUW. His sons with known descendants were Pieter Willemse;Dirk Willemse; and Teunis Willemse Van Slyke
I have primary sources as proof (wills, baptismal records, marriage records, West Indies Company account books aka passenger lists, Court records etc) and you can read more about this family on my webpage The Olive Tree Genealogy