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Saturday, October 1, 2016

Entry 32

It is well-known that1π=22992k=0(4k)!k!4297013k+1103(3964)kBut it turns out we can also use the square root 3964=±3962 as a median point for two other pi formulas. First express the above as1π=1922(3962)3/2k=0(4k)!k!425815015k+72798(3964)kthen1π=1922(396216)3/2k=0(2kk)kj=0(kj)(2k2jkj)(2jj)5815015k+(72798+37)(3962+16)k1π=1922(3962+16)3/2k=0(2kk)kj=0(kj)(2k2jkj)(2jj)5815015k+(7279837)(3962+16)k where (nk) is the binomial coefficient. Note that they have a beautifully symmetric form and how the same integers (which figure in Pell equations as discussed in Entry 1) appear in all three formulas. These two are level-8 Ramanujan-Sato series.

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