A062964 Pi 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, 3, 7, 0, 7, 3, 4, 4, 10, 4, 0, 9, 3, 8, 2, 2, 2, 9, 9, 15, 3, 1, 13, 0, 0, 8, 2, 14, 15, 10, 9, 8, 14, 12, 4, 14, 6, 12, 8, 9, 4, 5, 2, 8, 2, 1, 14, 6, 3, 8, 13, 0, 1, 3, 7, 7, 11, 14, 5, 4, 6, 6, 12, 15, 3, 4, 14, 9

Offset: 1

Author

Robert Lozyniak (11(AT)onna.com), Jul 22 2001

Keywords

• easy
• nonn
• base
• cons

Comments

Bailey and Crandall conjecture that the terms of this sequence, apart from the first, are given by the formula floor(16*(x(n) - floor(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 (see A374334). They have numerically verified the conjecture for the first 100000 terms of the sequence. - Peter Bala, Oct 31 2013

Bailey, Borwein & Plouffe's ("BBP") formula allows one to compute the n-th hexadecimal digit of Pi without calculating the preceding digits (see Wikipedia link). - M. F. Hasler, Mar 14 2015

Examples

			3.243f6a8885a308d3...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 17-28.

Links

• Harry J. Smith, Table of n, a(n) for n = 1..20000
• D. H. Bailey, Compendium of BBP-Type Formulas for Mathematical Constants.
• D. H. Bailey and R. E. Crandall, On the Random Character of Fundamental Constant Expansions, Experiment. Math. Volume 10, Issue 2 (2001), 175-190.
• CalcCrypto, Pi in Hexadecimal. [Broken link]
• Steven R. Finch, The Miraculous Bailey-Borwein-Plouffe Pi Algorithm.
• Steve Pagliarulo, Stu's pi page: base 16 (31 pages of numbers). [Dead link]
• Johnny Vogler, More digits.
• Wikipedia, Bailey-Borwein-Plouffe formula.

Crossrefs

Pi in base b: A004601 (b=2), A004602 (b=3), A004603 (b=4), A004604 (b=5), A004605 (b=6), A004606 (b=7), A006941 (b=8), A004608 (b=9), A000796 (b=10), A068436 (b=11), A068437 (b=12), A068438 (b=13), A068439 (b=14), A068440 (b=15), this sequence (b=16), A060707 (b=60).

Cf. A007514, A374334.

Programs

Mathematica

RealDigits[ N[ Pi, 115], 16] [[1]]

PARI

{ default(realprecision, 24300); x=Pi; for (n=1, 20000, d=floor(x); x=(x-d)*16; write("b062964.txt", n, " ", d)); } \\ Harry J. Smith, Apr 27 2009

PARI

N=50; default(realprecision,.75*N); A062964=digits(Pi*16^N\1,16) \\ M. F. Hasler, Mar 14 2015

Formula

a(n) = 8*A004601(4n) + 4*A004601(4n+1) + 2*A004601(4n+2) + 1*A004601(4n+3).

If Pi is the expansion of Pi in base 10, Pi=3.1415926...: a(n) = floor(16^n*Pi) - 16*floor(16^(n-1)*Pi). - Benoit Cloitre, Mar 09 2002

Extensions

More terms from Henry Bottomley, Jul 24 2001