Today, the flags of Ohio and Nebraska fly in commemoration of their admission as the seventeenth and thirty seventh states, respectively.
Ohio

Adopted on May 9, 1902, the Ohio Burgee is the only non-rectangular state flag in the United States. It’s a swallowtail flag, described in Ohio Revised Code § 5.01 as follows: “The flag of the state shall be burgee-shaped. It shall have three red and two white horizontal stripes that represent the roads and waterways of the state. The union of the flag shall be seventeen five-pointed stars, white in a blue triangular field that represents the state’s hills and valleys, the base of which shall be the staff end or vertical edge of the flag, and the apex of which shall be the center of the middle red stripe. The stars shall be grouped around a red disc superimposed upon a white circular “O.” The thirteen stars grouped around the “O” represent the original states of the United States and the four stars added to the peak of the triangle symbolize that Ohio was the seventeenth state admitted to the union. The “O” represents the “O” in “Ohio” and suggests the state’s nickname, the buckeye state. The proportional dimensions of the flag and of its various parts shall be according to the official design on file in the office of the secretary of state. One state flag of uniform dimensions shall be furnished to each company of the organized militia.”
Nebraska

Nebraska Revised Statues §90-102 defines the flag as follows, “The banner of the State of Nebraska shall consist of a reproduction of the Great Seal of the State, charged on the center in gold and silver on a field of national blue. The banner shall be the official state flag of the State of Nebraska and may be displayed on such occasions, at such times, and under such conditions as the flag of the United States of America. The banner shall be displayed on or near the State Capitol, the Governor’s Mansion, all courthouses, city or village halls, schoolhouses, and other public administrative buildings in this state under or to the left of the flag of the United States of America.”
Designed by Florence Hazen Miller, it was officially adopted as the state banner on April 2, 1925 and as the state flag on July 16, 1963.