A118031 Decimal expansion of the sum of the reciprocals of the palindromic numbers A002113.

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, 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, 1, 9, 7, 6, 0, 3, 4, 9, 4, 7, 6, 7, 5, 8, 9, 7, 0, 3, 0, 1

Offset

1,1

Comments

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

Links

Joseph Myers, Table of n, a(n) for n = 1..1001

Joseph Myers, Polynomial-time algorithm.

Michael Penn, Does this series converge??, YouTube video, 2021.

Radovan Potůček, Formulas for the Sums of the Series of Reciprocals of the Polynomial of Degree Two with Non-zero Integer Roots, Algorithms as a Basis of Modern Applied Mathematics, Studies in Fuzziness and Soft Computing book series (STUDFUZZ, Vol. 404) Springer (2021), 363-382.

Eric Weisstein's World of Mathematics, Palindromic Number.

Formula

a(n) = Sum_{palindromes p>0} 1/p.

a(n) = Sum_{n>=2} 1/A002113(n).

Example

3.3702832594973733204921572985...

Mathematica

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 *)

Crossrefs

Cf. A002113.

Similar sequences: A118064, A194097, A244162.

Sequence in context: A366820 A066358 A261295 * A335726 A240504 A338775

Adjacent sequences: A118028 A118029 A118030 * A118032 A118033 A118034

Keyword

cons,base,nonn

Author

Martin Renner, May 11 2006

Extensions

Corrected by Eric W. Weisstein, May 14 2006

Corrected and extended by Robert G. Wilson v, Oct 20 2010

Corrected and extended by Joseph Myers, Jun 26 2014

Status

approved