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%I #44 Jan 02 2023 12:30:46
%S 3,3,7,0,2,8,3,2,5,9,4,9,7,3,7,3,3,2,0,4,9,2,1,5,7,2,9,8,5,0,5,5,3,1,
%T 1,2,3,0,7,1,4,5,7,7,7,9,4,5,2,7,7,8,4,9,1,3,3,5,0,6,8,9,2,5,9,8,2,5,
%U 1,9,7,6,0,3,4,9,4,7,6,7,5,8,9,7,0,3,0,1
%N Decimal expansion of the sum of the reciprocals of the palindromic numbers A002113.
%C The sum using all palindromic numbers < 10^8 is 3.37000183240... Extrapolating using Wynn's epsilon method gives a value near 3.37018... - _Eric W. Weisstein_, May 14 2006
%H Joseph Myers, <a href="/A118031/b118031.txt">Table of n, a(n) for n = 1..1001</a>
%H Joseph Myers, <a href="http://list.seqfan.eu/oldermail/seqfan/2014-June/013186.html">Polynomial-time algorithm</a>.
%H Michael Penn, <a href="https://www.youtube.com/watch?v=YrO9OjTFSQs">Does this series converge??</a>, YouTube video, 2021.
%H Radovan Potůček, <a href="https://doi.org/10.1007/978-3-030-61334-1_18">Formulas for the Sums of the Series of Reciprocals of the Polynomial of Degree Two with Non-zero Integer Roots</a>, Algorithms as a Basis of Modern Applied Mathematics, Studies in Fuzziness and Soft Computing book series (STUDFUZZ, Vol. 404) Springer (2021), 363-382.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PalindromicNumber.html">Palindromic Number</a>.
%F a(n) = Sum_{palindromes p>0} 1/p.
%F a(n) = Sum_{n>=2} 1/A002113(n).
%e 3.3702832594973733204921572985...
%t NextPalindrome[n_] := Block[{l = Floor[ Log[10, n] + 1], idn = IntegerDigits@ n}, If[ Union@ idn == {9}, Return[n + 2], If[l < 2, Return[n + 1], If[ FromDigits[ Reverse[ Take[ idn, Ceiling[l/2]]]] > FromDigits[ Take[idn, -Ceiling[l/2]]], FromDigits[Join[Take[idn, Ceiling[l/2]], Reverse[Take[idn, Floor[l/2]]]]], idfhn = FromDigits[Take[idn, Ceiling[l/2]]] + 1; idp = FromDigits[ Join[ IntegerDigits@ idfhn, Drop[ Reverse[ IntegerDigits@ idfhn], Mod[l, 2]]]]]]]]; pal = 1; sm = 0; Do[ While[pal < 10^n + 1, sm = N[sm + 1/pal, 128]; pal = NextPalindrome@ pal]; Print[{n, sm}], {n, 0, 17}] (* _Robert G. Wilson v_, Oct 20 2010 *)
%Y Cf. A002113.
%Y Similar sequences: A118064, A194097, A244162.
%K cons,base,nonn
%O 1,1
%A _Martin Renner_, May 11 2006
%E Corrected by _Eric W. Weisstein_, May 14 2006
%E Corrected and extended by _Robert G. Wilson v_, Oct 20 2010
%E Corrected and extended by _Joseph Myers_, Jun 26 2014
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