[seqfan] Re: Palindome reciprocal sums
Hans Havermann
gladhobo at teksavvy.com
Mon Jun 23 13:53:29 CEST 2014
A convenient base-b extrapolation shortcut is to take the calculated value at 2n+1 binary digits, multiply it by b, subtract the value at 2n-1, and divide the result by (b-1). In base two: twice the value at 2n+1 minus the value at 2n-1.
57 2.37879570237701206804291577 and
59 2.37879570495918653518910877 with only 9 of the sequence's digits
yields 2.37879570754136100233530177 with 25 terms!
In base ten:
15 3.3702832083288156984016021 and
17 3.3702832543805175582831661 with 9 of A118031's digits
yields 3.3702832594973733204922288 with 22 terms.
The number of significant terms may be guessed from the approach:
11 => 3.3702832594974151638818129
13 => 3.3702832594973733722281115
15 => 3.3702832594973733205537858
17 => 3.3702832594973733204922288
Of course, now we can check them directly against Joseph Myers' deeply exact value. :)
On Jun 22, 2014, at 5:33 PM, M. F. Hasler wrote:
> Aitken applied to every other term of that sequence yields the constant sequence 2.3787957075414...
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