The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A072691 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A072691

Decimal expansion of Pi^2/12.
(history; published version)

#140 by Peter Luschny at Thu Aug 22 05:16:48 EDT 2024

STATUS

reviewed

approved

#139 by Vaclav Kotesovec at Thu Aug 22 04:37:16 EDT 2024

STATUS

proposed

reviewed

Discussion

Thu Aug 22

05:26

Peter Bala: @ Vaclav: Yes, the formula is correct - the proof is in the Catalan link (Mma giving a1000 d.p. match is, of course, not a proof). The point about taking only 10 terms of the series and getting 10 d.p. accuracy was to indicate to the reader that the series was rapidly converging. Can you restore my comment.

#138 by Vaclav Kotesovec at Thu Aug 22 04:37:10 EDT 2024

STATUS

editing

proposed

#137 by Vaclav Kotesovec at Thu Aug 22 04:36:56 EDT 2024

FORMULA

Equals 1 + (1/8)*Sum_{k >= 0} (-1)^(k-1) * (10*k + 13)/((k + 1)*(2*k + 1)^2*(2*k + 3)^2*binomial(2*k, k)). See Catalan, Section 35, equation 54. Taking 10 terms of the series gives Pi^2/12 = 0.8224670334(19...), correct to 10 decimal places. - Peter Bala, Aug 17 2024

STATUS

proposed

editing

#136 by Amiram Eldar at Wed Aug 21 16:17:50 EDT 2024

STATUS

editing

proposed

Discussion

Wed Aug 21

17:17

Peter Luschny: Yes. URL is the 'locator' not the location. In fact, I immediately added your 'link' to A006752.

Thu Aug 22

04:15

Peter Bala: Thanks Amiram.

04:36

Vaclav Kotesovec: Formula is correct. Mathematica gives 1 + 1/360 (-1800 + 1800 Sqrt[5] ArcSinh[1/2] - 
    130 HypergeometricPFQ[{1/2, 1, 1}, {5/2, 5/2}, -(1/4)] + 
    45 HypergeometricPFQ[{1, 1, 1}, {2, 5/2}, -(1/4)] - 
    130 HypergeometricPFQ[{1, 1, 3/2}, {5/2, 5/2}, -(1/4)] + 
    2 HypergeometricPFQ[{3/2, 2, 2}, {7/2, 7/2}, -(1/4)] + 
    6 HypergeometricPFQ[{2, 2, 5/2}, {7/2, 7/2}, -(1/4)]), match to 1000 or more decimal places.

#135 by Amiram Eldar at Wed Aug 21 16:17:08 EDT 2024

LINKS

Eugène-Charles Catalan, <a href="https://www.persee.fr/doc/marb_0776-3123_1867_num_33_1_1705">Mémoire sur la transformation des séries et sur quelques intégrales définies<\/a>, Mémoires de l'Académie royale de Belgique, 1867, Vol. 33, pp. 1-50.

Discussion

Wed Aug 21

16:17

Amiram Eldar: The problem was not with the link, but with the formatting. I have fixed it.

#134 by Alois P. Heinz at Wed Aug 21 15:38:13 EDT 2024

STATUS

proposed

editing

Discussion

Wed Aug 21

15:38

Alois P. Heinz: click blue #134 above and see!

#133 by Peter Bala at Wed Aug 21 15:23:13 EDT 2024

STATUS

editing

proposed

#132 by Peter Luschny at Mon Aug 19 13:53:08 EDT 2024

STATUS

proposed

editing

Discussion

Wed Aug 21

15:23

Peter Bala: I'm puzzled. The link works for me. It works fine in A076788.

#131 by Peter Bala at Mon Aug 19 07:48:04 EDT 2024

STATUS

editing

proposed

Discussion

Mon Aug 19

13:53

Peter Luschny: url is not ok.