%I #70 Mar 11 2023 05:22:46

%S 31,26,93,62,82,28,28,28,48,48,48,48,48,9,9,81,17,17,95,95,95,95,95,

%T 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 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 First 2-digit number to appear n times in the decimal expansion of Pi.

%C 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 Note that the sequence contains elements whose number of digits is 2 or 1, see examples. - _Omar E. Pol_, Oct 05 2016

%C Comment from _N. J. A. Sloane_, Mar 08 2023 (Start)

%C Make a table T[0,0], T[0,1], ...,T[9,9], with 100 columns, labeled 0,0 to 9,9.

%C Scan the digits of pi = 3.14159....

%C 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 The first time any count hits 6, say T[3,8] = 6, then a(6) = 38.

%C 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 (End)

%C Comment from _Alois P. Heinz_, Mar 08 2023 (Start)

%C 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 In the first 10000 terms we see "09" 40 times, "14" 33 times, and so on. Here is the complete list:

%C [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 67 of the two-digit strings never show up in the first 10000 terms.

%C 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 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 William Feller, An Introduction to Probability Theory and Its Applications, Vol. I, Chapter III, Wiley, 3rd Ed., Corrected printing 1970.

%H Alois P. Heinz, <a href="/A276992/b276992.txt">Table of n, a(n) for n = 1..10000</a>

%e 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 a(14) = 9 because "09" is the first 2-digit number to appear 14 times in the decimal expansion of Pi.

%t 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 Cf. A000796, A096567, A276686, A276993, A277171, A277270, A291599, A291600.

%K nonn,base

%O 1,1

%A _Bobby Jacobs_, Sep 24 2016

%E a(21)-a(40) from _Bobby Jacobs_, Oct 01 2016

%E More terms from _Alois P. Heinz_, Oct 02 2016