

A104865


Length of sections with the same initial and final digits in the decimal expansion of Pi.


7



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, 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, 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
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OFFSET

1,1


COMMENTS

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,...
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},...
Then A104865(n) = length of [s(n)]: 10,22,19,11,...
A104866(n) = final digits of [s(n)]: 3,5,0,5,...
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.
A104869 gives integer values in sum[A104865(i),{i,1,n}]/n: 10,16,17,16,15,14,14,13,11,11,11,11,...
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,...


LINKS



MATHEMATICA

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 *)


PROG

(Python)
from sympy import S
# download https://stuff.mit.edu/afs/sipb/contrib/pi/pibillion.txt, then
# with open('pibillion.txt', 'r') as f: pi_digits = f.readline()
pi_digits = str(S.Pi.n(2*10**5))[:1] # alternative to above
pi_digits = pi_digits.replace(".", "")
def aupton(terms):
alst, idx = [], 0
for n in range(1, terms+1):
digit, idx, c = pi_digits[idx], idx+1, 2
while pi_digits[idx] != digit:
idx += 1; c += 1
assert idx < len(pi_digits), "increase precision"
alst.append(c); idx += 1
return alst


CROSSREFS



KEYWORD

nonn,base


AUTHOR



EXTENSIONS



STATUS

approved



