A382307 Position of start of first run of alternating bit values in the base-2 representation of Pi, or -1 if no such run exists.

1, 2, 2, 19, 19, 19, 19, 19, 1195, 1697, 1890, 1890, 1890, 1890, 15081, 63795, 206825, 206825, 206825, 470577, 470577, 557265, 557265, 557265, 557265, 557265, 447666572, 447666572, 699793337, 699793337, 2049646803, 2250772991

Offset

1,2

Comments

In base-2, Pi is: 11.00100100001111110110101010001... For this sequence, the integer part of Pi is ignored, and the first fractional bit is numbered one.

The bit substring may begin with either 0 or 1.

a(27) > 4*10^6. - Michael S. Branicky, Mar 23 2025

Conjecture: no term is -1 (would follow from a proof that Pi is normal in base 2). - Sean A. Irvine, Mar 30 2025

a(33) > 4*10^9. - Pontus von Brömssen, Apr 06 2025

Links

Table of n, a(n) for n=1..32.

Index entries for sequences related to the number Pi.

Example

The first alternating bit run is "0", at position 1, so a(1) = 1. The second and third alternating bit runs are "01" and "010", starting at position 2, so a(2) and a(3) are both 2.
a(4)-a(8) = 19 since the binary digits of Pi are "10101010" starting at position 19.

Prog

(Python)
def binary_not(binary_string):
    return ''.join('1' if bit == '0' else '0' for bit in binary_string)
# !pip install gmpy2   # may be necessary
import gmpy2
gmpy2.get_context().precision = 12000000
pi = gmpy2.const_pi()
# Convert Pi to binary representation
binary_pi = gmpy2.digits(pi, 2)[0] # zero-th element is the string of bits
binary_pi = binary_pi[2:] # remove leading "11" left of decimal point
outVec = []
strSearch0 = "" # this substring starts with "0"
for lenRun in range(1, 30):
  strSearch0 += "0" if lenRun%2==1 else "1"
  strSearch1 = binary_not(strSearch0)
  l0 = binary_pi.find(strSearch0)+1 # incr origin-0 result
  l1 = binary_pi.find(strSearch1)+1
  outVec.append(min(l0, l1))
print(outVec)

Crossrefs

Cf. A004601, A175945, A178708, A233836, A378472.

Sequence in context: A318418 A184317 A141084 * A172194 A338400 A093777

Adjacent sequences: A382304 A382305 A382306 * A382308 A382309 A382310

Keyword

nonn,base,more

Author

James S. DeArmon, Mar 21 2025

Extensions

a(26)-a(32) from Pontus von Brömssen, Apr 06 2025

Status

approved