Template:Sequence of the Day for March 14

Intended for: March 14, 2016 ( 

π

day)

Timetable

• First draft entered by M. F. Hasler on March 14, 2015 ✓
• Draft reviewed by Daniel Forgues on March 11, 2016, March 13, 2018 ✓
• Draft to be approved by February 14, 2016

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A062964:

π

(referred to as “Pi” in the OEIS) in hexadecimal.

{ 3, 2, 4, 3, 15, 6, 10, 8, 8, 8, 5, 10, 3, 0, 8, 13, 3, 1, 3, 1, 9, 8, 10, 2, 14, 0, ... }

The use of base 16 is very common for calculations in machines using binary arithmetic. (The 16 hexadecimal digits, usually from

{0, ..., 9, A, ..., F}

, correspond to “nibbles” or even “nybbles,” i.e. half “bytes,” to match the vowels of byte, and bytes are made of two nibbles,

00 –FF

.) It happens that base 16 is also very relevant for calculation of digits of

π

. Indeed, Bailey, Borwein and Plouffe have found the formula

π  =    ∞ ∑   k  = 0    [ 1 16 k ( 4 8k + 1 − 2 8k + 4 − 1 8k + 5 − 1 8k + 6 )],

referred to as the BPP formula, which allows to compute a given base-16 digit of

π

without calculating the preceding digits.
Bailey and Crandall conjecture that the terms of this sequence, apart from the first (which is the integer part), are given by the formula

⌊  16 ( x (n)  −  ⌊  x (n)⌋  )⌋

, where

x (n)

is determined by the recurrence equation

x (n)  =  16 x  (n − 1) + 120 n 2 − 89 n + 16 512 n 4 − 1024 n 3 + 712 n 2 − 206 n + 21  ,

with the initial condition

x (0) = 0

. They have numerically verified the conjecture for the first 100,000 terms of the sequence.

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