You are given the sequence:
$$\frac{1}{4}, 1, \frac{9}{4}, 2 + \frac{2}{\sqrt{3}}, \frac{5}{2} + \frac{5}{2 \sqrt{2}}, 3 + 3 \sqrt{\frac{1}{10} (5 + \sqrt{5})}$$
Transcription: one fourth, then one, then nine fourths, then two plus the fraction two over the square root of three, then five halves plus the fraction five over the product of two and the square root of two, then three plus the product of three and the square root of the product of one tenth and the sum of five and the square root of five
What is the next number? Explain
Hint: The next number is rational.
Explanation hint: this is a series of coefficients of a constant. To be clear, I mean that there exists a constant which, if multiplied by each these numbers, gives the actual values of interest. Chris gives a helpful example: "a sequence which is the area of circles with radius n would be pi, 4pi, 9pi, etc. The coefficients of the constant would just be 1, 4, 9, etc" But pi is not the correct constant for this particular sequence.
New hint: the correct constant is pi^2
This is my own puzzle