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%I #12 Oct 22 2021 23:56:32
%S 10,22,19,11,18,21,5,11,2,7,20,14,6,44,4,34,17,10,6,15,8,12,10,4,11,
%T 13,21,24,16,5,11,17,19,39,33,17,4,8,7,3,20,10,6,4,21,20,11,12,3,5,4,
%U 5,27,2,3,21,7,22,13,7,6,8,4,4,8,2,2,8,4,4,11,3,9,28,7,49,30,3,5,8,24,5,11,3
%N Length of sections with the same initial and final digits in the decimal expansion of Pi.
%C Start with decimal expansion of Pi: s0=3,1,4,1,5,9,2,6,5,3,5,8,9,7,9,3,2,3,8,4,6,2,6,4,3,3,8,3,2,7,9,5,0,2,8,8,4,1,9,7,1,6,9,3,9,9,3,7,5,1,0,5,8,2,0,9,7,4,9,4,4,5,...
%C Divide s0 to the sections with the same final digits: s={3,1,4,1,5,9,2,6,5,3}, {5,8,9,7,9,3,2,3,8,4,6,2,6,4,3,3,8,3,2,7,9,5}, {0,2,8,8,4,1,9,7,1,6,9,3,9,9,3,7,5,1,0}, {5,8,2,0,9,7,4,9,4,4,5},...
%C Then A104865(n) = length of [s(n)]: 10,22,19,11,...
%C A104866(n) = final digits of [s(n)]: 3,5,0,5,...
%C A104867 gives records in A104865: 10,22,44,49,57,70,...
%C A104871 gives positions of records A104867 in A104865: 1,2,14,76,146,311;
%C A104868(n) = round value of the running average of A104865(n)= A104868(n) = round(sum[A104865(i),{i,1,n}]/n)= 10,16,17,16,16,17,15,15,13,... A104868(n) => 11 as n => Infinity.
%C A104869 gives integer values in sum[A104865(i),{i,1,n}]/n: 10,16,17,16,15,14,14,13,11,11,11,11,...
%C A104870 gives values of n such that sum[A104865(i),{i,1,n}]/n is integer: 1,2,3,5,14,30,32,63,174,216,312,324,...
%t pi = RealDigits[Pi, 10, 1000][[1]]; lst = {}; f := Block[{k = Position[pi, pi[[1]]][[2, 1]]}, pi = Drop[pi, k]; k]; Do[lst = {lst, f}, {n, 80}]; Flatten[lst] (* _Robert G. Wilson v_, Mar 29 2005 *)
%o (Python)
%o from sympy import S
%o # download https://stuff.mit.edu/afs/sipb/contrib/pi/pi-billion.txt, then
%o # with open('pi-billion.txt', 'r') as f: pi_digits = f.readline()
%o pi_digits = str(S.Pi.n(2*10**5))[:-1] # alternative to above
%o pi_digits = pi_digits.replace(".", "")
%o def aupton(terms):
%o alst, idx = [], 0
%o for n in range(1, terms+1):
%o digit, idx, c = pi_digits[idx], idx+1, 2
%o while pi_digits[idx] != digit:
%o idx += 1; c += 1
%o assert idx < len(pi_digits), "increase precision"
%o alst.append(c); idx += 1
%o return alst
%o print(aupton(84)) # _Michael S. Branicky_, Oct 22 2021
%Y Cf. A000796, A104866, A104867, A104868, A104869, A104870, A104871.
%K nonn,base
%O 1,1
%A _Zak Seidov_, Mar 29 2005
%E More terms from _Robert G. Wilson v_, Mar 29 2005
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