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A119744 a(n) is the position of a(n-1) in the decimal expansion of Pi, starting with a(1)=13.

%I A119744 #9 Sep 13 2024 06:51:34
%S A119744 13,111,154,3645,2699,1394,526,613,971,39,44,60,128,149,2607,23047,
%T A119744 21876,361554,403706,1674698,19210577
%N A119744 a(n) is the position of a(n-1) in the decimal expansion of Pi, starting with a(1)=13.
%H A119744 Dave Andersen, <a href="http://www.angio.net/pi/piquery">The Pi-Search Page</a>
%e A119744 In the decimal expansion of Pi (A000796) written as a string
%e A119744 3141592653589793238462643383279502884197169399375105820974944592307816...,
%e A119744 the string "13" is found at position 111, the string "111" is at position 154, the string "154" at position 3645, etc.,
%e A119744 hence the sequence starting with a(1)=13 is
%e A119744 13,111,154,3645,2699,...
%e A119744 In general, the sequence may end in a cycle, e.g. sequence s1
%e A119744 starting with a(1)=1 is
%e A119744 s1: 1,2,7,14,2,7,14,2,7,14,2 (cycle is 2,7,14),
%e A119744 Also s0, s2, s3, s4, s7, s10, s11, s14, s15, s16, s25 end with the same cycle:
%e A119744 s0: 0,33,25,90,248,480,105,50,32,16,41,3,1,2,7,14,2
%e A119744 s2: 2,7,14,2,7,14,2,
%e A119744 s3: 3,1,2,7,14,2,7,14,2
%e A119744 s4: 4,3,1, 2,7,14,2,7,14,2
%e A119744 s5: 5,5,5,5 (simplest cycle!)
%e A119744 s6: 6, 8, 12,149,2607,23047,21876,361554,403706,1674698,19210577,next term>2*10^8
%e A119744 s7: 7,14,2,7, 14, (see s1)
%e A119744 s8: 8,12,149, (see s6)
%e A119744 s9: 9,6, 8, (see s6)
%e A119744 s10: 10,50,32,16,41,3,1,2, (see s1)
%e A119744 s11: 11,95,31,1, (see s1)
%e A119744 s12: 12,149, (see s6)
%e A119744 s13: this sequence, is there cycle or not? next term>2*10^8
%e A119744 s14: 14,2, (see s1)
%e A119744 s15: 15,4,3,1, (see s1)
%e A119744 s16: 16,41,3,1, (see s1)
%e A119744 s17: 17,96,181,729,771,626,21,94,59,5,5,5,(see s5)
%e A119744 s18: 18,425,822,135,2728,11023,12721,54517,102917,183252,410024,613425,1525497,
%e A119744 3426169,3591590,10748112, is there cycle or not? next term>2*10^8
%e A119744 s19: 19,38,18, (see s18), is there cycle or not? next term>2*10^8
%e A119744 s20: 20,54,192,976,1808,26035,43352,93226,3603,9736,10514,54423,140517,1549413,
%e A119744 20801035, is there cycle or not? next term>2*10^8
%e A119744 s21: 21,94,59,5,5,5,(see s5)
%e A119744 s22: 22,136,735,469,387,864,722,2140,8434,9666,4000,14637,85171,3538,5037,37934,
%e A119744 62186,6529,37803,68887,5871,22098,172393,591481,14933,51852, 5762,7347,11749,
%e A119744 12529,61828,268516,657761,531469,1246616,6755774,22119206,83934772,128149562,
%e A119744 is there cycle or not? next term>2*10^8
%e A119744 s23: 23,17,96,181,729,771,626,21,94,59,5,5,5, (see s17, s5)
%e A119744 s24: 24,293,572,405,596,180,3665,10143,63892,465223,522194,1637321,10980764,
%e A119744 184160876,65620598,35543320,97248583,109914084,40782089,
%e A119744 48875829,77976212,182755461,114041877, is there cycle or not? next term>2*10^8
%e A119744 s25: 25,90,248,480,105,50,32,16,41,3,1,2,7,14,2 (see s0, s1)
%Y A119744 Cf. A000796 = Decimal expansion of Pi, A097614 = sequence based on positions of digits in decimal digits of Pi.
%K A119744 base,nonn,more
%O A119744 1,1
%A A119744 _Zak Seidov_, Jun 16 2006
%E A119744 Edited by _N. J. A. Sloane_, Dec 09 2017