A104866 Initial and final digits in the partition of decimal expansion of Pi.

3, 5, 0, 5, 9, 9, 8, 0, 6, 4, 4, 0, 2, 7, 5, 9, 0, 3, 2, 1, 4, 5, 5, 2, 0, 9, 2, 0, 9, 5, 3, 8, 5, 6, 7, 3, 7, 7, 1, 7, 5, 1, 2, 6, 0, 9, 0, 1, 0, 7, 9, 2, 3, 4, 1, 5, 1, 1, 0, 9, 5, 9, 0, 2, 5, 3, 4, 6, 1, 1, 7, 8, 7, 5, 9, 0, 5, 9, 2, 0, 9, 3, 3, 0, 1, 8, 9, 5, 9, 4, 9, 1, 5, 5, 7, 2, 5, 0, 4, 8

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) = initial/final digits of s(n): 3,5,0,5,...

Links

Table of n, a(n) for n=1..100.

Mathematica

pi=First@RealDigits[Pi, 10, 10^6]; lst={pi[[1]]}; k=q=1; Do[While[pi[[++k]]!=lst[[q]]]; AppendTo[lst, pi[[k+1]]]; q++; k++, 100]; lst (* Giorgos Kalogeropoulos, Oct 22 2021 *)

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(int(digit)); idx += 1
    return alst
print(aupton(100)) # Michael S. Branicky, Oct 22 2021

Crossrefs

Cf. A000796, A104865, A104867, A104868, A104869, A104870, A104871.

Sequence in context: A243967 A144541 A100609 * A165723 A257407 A224932

Adjacent sequences: A104863 A104864 A104865 * A104867 A104868 A104869

Keyword

nonn,base

Author

Zak Seidov, Mar 29 2005

Status

approved