# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a276992 Showing 1-1 of 1 %I A276992 #70 Mar 11 2023 05:22:46 %S A276992 31,26,93,62,82,28,28,28,48,48,48,48,48,9,9,81,17,17,95,95,95,95,95, %T A276992 95,95,19,21,21,21,19,95,9,9,9,95,46,95,59,9,9,9,95,95,95,95,59,59,59, %U A276992 9,9,9,9,9,9,9,9,9,9,9,9,9,9,14,14,14,9,9,9,9,14,9,9 %N A276992 First 2-digit number to appear n times in the decimal expansion of Pi. %C A276992 a(n) is the 2-digit number that appears in Pi n times before any other 2-digit number appears in Pi n times. %C A276992 Note that the sequence contains elements whose number of digits is 2 or 1, see examples. - _Omar E. Pol_, Oct 05 2016 %C A276992 Comment from _N. J. A. Sloane_, Mar 08 2023 (Start) %C A276992 Make a table T[0,0], T[0,1], ...,T[9,9], with 100 columns, labeled 0,0 to 9,9. %C A276992 Scan the digits of pi = 3.14159.... %C A276992 First you see 3, 1 so increment the count for 3,1; next you see 1,4, so increment the count for 1,4. Then you see 4,1 so increment the count for 4,1. Do this for ever. %C A276992 The first time any count hits 6, say T[3,8] = 6, then a(6) = 38. %C A276992 If it happens that T[0,9] hits 6 first, then a(6) would be 09, but we would drop the 0, and write a(6) = 9. %C A276992 (End) %C A276992 Comment from _Alois P. Heinz_, Mar 08 2023 (Start) %C A276992 Initially, "09" is very often the first to occur n times, while other 2-digit substrings fall behind. They can show up later. This is not strange, this is Pi. %C A276992 In the first 10000 terms we see "09" 40 times, "14" 33 times, and so on. Here is the complete list: %C A276992 [40, 9], [33, 14], [2, 17], [13, 19], [3, 21], [1, 26], [892, 27], [3, 28], [1, 31], [144, 34], [107, 35], [179, 39], [2594, 46], [5, 48], [127, 54], [1387, 55], [4, 59], [6, 62], [41, 65], [671, 71], [19, 74], [3406, 76], [1, 81], [1, 82], [94, 85], [1, 93], [211, 94], [14, 95]. %C A276992 67 of the two-digit strings never show up in the first 10000 terms. %C A276992 It does not mean that they do not appear in Pi. Indeed they do. It only means that they are never the first to reach some count. They may be behind by only a small amount. (End) %C A276992 The fact that 09 is ahead so often is an example of the Arcsine Law Paradox at work. See for example Feller, Volume I, Chapter III. As Feller says, "[the conclusions] are not only unexpected but actually come as a shock to intuition and common sense." Of course the same phenomenon occurs with single digits of Pi, see A096567, where 5 seems to be ahead most of the time. - _N. J. A. Sloane_, Mar 09 2023 %D A276992 William Feller, An Introduction to Probability Theory and Its Applications, Vol. I, Chapter III, Wiley, 3rd Ed., Corrected printing 1970. %H A276992 Alois P. Heinz, Table of n, a(n) for n = 1..10000 %e A276992 a(2) = 26 because 26 is the first 2-digit number to appear 2 times in the decimal expansion of Pi = 3.14159(26)5358979323846(26)... %e A276992 a(14) = 9 because "09" is the first 2-digit number to appear 14 times in the decimal expansion of Pi. %t A276992 spi = ToString[Floor[10^100000 Pi]]; f[n_] := Block[{k = 2}, While[Length@ StringPosition[ StringTake[spi, k], StringTake[spi, {k - 1, k}]] != n, k++]; ToExpression@ StringTake[spi, {k - 1, k}]]; Apply[f, 72] (* _Robert G. Wilson v_, Oct 05 2016 *) %Y A276992 Cf. A000796, A096567, A276686, A276993, A277171, A277270, A291599, A291600. %K A276992 nonn,base %O A276992 1,1 %A A276992 _Bobby Jacobs_, Sep 24 2016 %E A276992 a(21)-a(40) from _Bobby Jacobs_, Oct 01 2016 %E A276992 More terms from _Alois P. Heinz_, Oct 02 2016 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE