Recurrence sequence in decimal expansion of Pi.

[zak seidov]

Dear seqfans,
just submitted,
sorry for lengthy text,
in Neil's leave,
it'd be great if someone'll be interested:

a) is there cycle for ANY a(1)?
b) what is the longest sequence with a(n)<2*10^8
(limit in Dave Andersen's site)?
(* shortest is s5: 5,5,5,5 *)
c) is it possible to push farther this limit?

thanks, Zak

   
%I A000001
%S A000001 
13,111,154,3645,2699,1394,526,613,971,39,44,60,128,149,2607,23047,21876,361554,403706,1674698,19210577
%N A000001 a(n) is the position of a(n-1) in the
decimal expansion of 
Pi,  a(1)=13.
%C A000001 A000796 Decimal expansion of Pi, A097614 
Recurrence 
sequence based on positions of digits in decimal
places of Pi.
%H A000001 Dave Andersen, <a 
href="http://www.angio.net/pi/piquery">The Pi-Search
Page</a>
%e A000001 In the decimal expansion of P A000796 
written down as 
string:
3141592653589793238462643383279502884197169399375105820974944592307816
the string "13" is found at position 111, the string
"111" is at 
position 154, the string "154" at position 3645, etc.,

hence the sequence starting with a(1)=13 is 
13,111,154,3645,2699,... 
In general, the sequence may end in cycle, e.g.
sequence s1
starting with a(1)=1 is 
s1: 1,2,7,14,2,7,14,2,7,14,2 (cycle 2,7,14,2),
Also s0, s2, s3, s4, s7, s10, s11, s14, s15, s16, s25
end with the same 
cycle:
s0: 0,33,25,90,248,480,105,50,32,16,41,3,1,2,7,14,2 
s2: 2,7,14,2,7,14,2,
s3: 3,1,2,7,14,2,7,14,2
s4: 4,3,1, 2,7,14,2,7,14,2
s5: 5,5,5,5 (simplest cycle!) 
s6: 6, 8,
12,149,2607,23047,21876,361554,403706,1674698,19210577,next

term>2*10^8
s7: 7,14,2,7, 14, (see s1)
s8: 8,12,149, (see s6)
s9: 9,6, 8, (see s6)
s10: 10,50,32,16,41,3,1,2, (see s1)
s11: 11,95,31,1, (see s1)
s12: 12,149, (see s6)
s13: this sequence, is there cycle or not? next
term>2*10^8
s14: 14,2, (see s1)
s15: 15,4,3,1, (see s1)
s16: 16,41,3,1, (see s1)
s17: 17,96,181,729,771,626,21,94,59,5,5,5,(see s5)
s18: 
18,425,822,135,2728,11023,12721,54517,102917,183252,410024,613425,1525497,
3426169,3591590,10748112, is there cycle or not? next
term>2*10^8
s19: 19,38,18, (see s18}, is there cycle or not? next
term>2*10^8
s20: 
20,54,192,976,1808,26035,43352,93226,3603,9736,10514,54423,140517,1549413,
20801035, is there cycle or not? next term>2*10^8
s21: 21,94,59,5,5,5,(see s5)
s22: 
22,136,735,469,387,864,722,2140,8434,9666,4000,14637,85171,3538,5037,37934,
62186,6529,37803,68887,5871,22098,172393,591481,14933,51852,

5762,7347,11749,
12529,61828,268516,657761,531469,1246616,6755774,22119206,83934772,128149562,
is there cycle or not? next term>2*10^8    
s23: 23,17,96,181,729,771,626,21,94,59,5,5,5, (see
s17, s5)
s24: 
24,293,572,405,596,180,3665,10143,63892,465223,522194,1637321,10980764,
184160876,65620598,35543320,97248583,109914084,40782089,
48875829,77976212,182755461,114041877, is there cycle
or not? next 
term>2*10^8  
s25: 25,90,248,480,105,50,32,16,41,3,1,2,7,14,2 (see
s0, s1)
%Y A000001 A000796, A097614.
%O A000001 1
%K A000001 ,base,nonn,
%A A000001 Zak Seidov (zakseidov at yahoo.com), Jun 16
2006


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