%I #55 Jan 29 2020 04:40:33

%S 2,149,1925,13808,49703,2458886,9470345,186557267,523551503,

%T 191278379840,4368196101672

%N a(n) is the first position in the digit sequence 3,1,4,1,5,9,.... of Pi where the pattern "1,2,...,n" occurs (where position of the initial 3 is 1).

%C 1. We may never know if a(n) is defined for all n.

%C 2. We split up the digits of any number > 9 in the pattern, e.g., if n = 11, we search for the pattern "1,2,3,4,5,6,7,8,9,1,0,1,1".

%C 3. The pattern "1,2,3,4,5,6" does not occur before the 100,000th term in the digit sequence of Pi.

%C Two more terms a(6) and a(7) were found via the referenced Pi-Search link [Andersen], through which 100 million digits of Pi are currently available. - _Rick L. Shepherd_, Oct 10 2002

%C 200 million digits now available at Pi-Search page. - _Rick L. Shepherd_, Aug 06 2006

%C This sequence uses position = 1 for the initial digit 3 of Pi, while A121280(n) = a(n)-1 starts counting at 0, as does the "Pi search page" and sequences A035117, A050279 - A050287, A048940, A096755 - A096763. - _M. F. Hasler_, Mar 18 2017

%C a(10) > 2*10^9. - _M. F. Hasler_, Apr 13 2019

%C a(12) > 22*10^12. - _Dmitry Petukhov_, Jan 29 2020

%D Wacław Sierpiński, O stu prostych, ale trudnych zagadnieniach arytmetyki. Warsaw: PZWS, 1959, p. 32.

%H D. G. Andersen, <a href="http://www.angio.net/pi/piquery">The Pi-Search Page</a>

%H SubIdiom.com, <a href="http://www.subidiom.com/pi">Irrational numbers search engine: π = 3.14159...</a>. (Search within 2*10^9 digits, since at least 2009, maybe 2002.)

%H Peter Trüb, <a href="https://pi2e.ch/blog/2017/03/10/pi-digits-download/">22.4 trillion digits of pi</a>

%F a(n) = A121280(n) + 1. - _M. F. Hasler_, Apr 13 2019

%t p = ToString[N[Pi, 50000]/10]; t = {1, 12, 123, 1234, 12345}; g[n_] := StringPosition[p, ToString[n]][[1]][[1]] - 2; Table[g[t[[i]]], {i, 1, 5}]

%Y First occurrence of n times the same digit: A035117 (n '1's), A050281 (n '2's), A050282, A050283, A050284, A050286, A050287, A048940 (n '9's).

%Y First occurrence of exactly n times the same digit: A096755 (exactly n '1's), A096756, A096757, A096758, A096759, A096760, A096761, A096762, A096763 (exactly n '9's), A050279 (exactly n '0's).

%Y First occurrence of n: A176341; of concatenate(1,...,n): A121280 = A068987 - 1.

%Y Cf. A000796: Decimal expansion (or digits) of Pi.

%K nonn,base,more

%O 1,1

%A _Joseph L. Pe_, Apr 01 2002

%E More terms from _Rick L. Shepherd_, Oct 10 2002

%E a(8) from _Rick L. Shepherd_, Aug 06 2006

%E Additional term a(9), using subidiom search engine, from _M. F. Hasler_, Apr 13 2019

%E a(10)-a(11) from _Dmitry Petukhov_, Jan 16 2020