Some new formulas for π

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We show how to find series expansions for π of the form π=∑ₙ₌₀∞ S(n)/mnpnaⁿ, where S(n) is some polynomial in n (depending on m,p,a). We prove that there exist such expansions for m=8k, p=4k, a=(-4)ᵏ, for any k, and give explicit examples for such expansions for small values of m, p and a.