A104865 Length of sections with the same initial and final digits in the decimal expansion of Pi.

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

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,...

A104867 gives records in A104865: 10,22,44,49,57,70,...

A104871 gives positions of records A104867 in A104865: 1,2,14,76,146,311;

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

Table of n, a(n) for n=1..84.

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/pi-billion.txt, then
# with open('pi-billion.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
print(aupton(84)) # Michael S. Branicky, Oct 22 2021

Crossrefs

Cf. A000796, A104866, A104867, A104868, A104869, A104870, A104871.

Sequence in context: A080454 A226789 A110367 * A063555 A228010 A303745

Adjacent sequences: A104862 A104863 A104864 * A104866 A104867 A104868

Keyword

nonn,base

Author

Zak Seidov, Mar 29 2005

Extensions

More terms from Robert G. Wilson v, Mar 29 2005

Status

approved