A062964 - OEIS

%I #67 Aug 30 2024 19:26:06

%S 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,

%T 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,

%U 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

%N Pi in hexadecimal.

%C 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

%C 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

%D S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 17-28.

%H Harry J. Smith, <a href="/A062964/b062964.txt">Table of n, a(n) for n = 1..20000</a>

%H D. H. Bailey, <a href="http://crd.lbl.gov/~dhbailey/dhbpapers/bbp-formulas.pdf">Compendium of BBP-Type Formulas for Mathematical Constants</a>.

%H D. H. Bailey and R. E. Crandall, <a href="https://projecteuclid.org/euclid.em/999188630">On the Random Character of Fundamental Constant Expansions</a>, Experiment. Math. Volume 10, Issue 2 (2001), 175-190.

%H CalcCrypto, <a href="http://calccrypto.wikidot.com/math:pi-hex">Pi in Hexadecimal</a>. [Broken link]

%H S. R. Finch, <a href="http://www.plouffe.fr/simon/articles/Miraculous.pdf">The Miraculous Bailey-Borwein-Plouffe Pi Algorithm</a>.

%H Steve Pagliarulo, <a href="http://members.shaw.ca/francislyster/pi/pistats/pibase16.pdf">Stu's pi page: base 16</a> (31 pages of numbers). [Dead link]

%H Johnny Vogler, <a href="http://www.math.byu.edu/~jvogler/pi.html">More digits</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Bailey%E2%80%93Borwein%E2%80%93Plouffe_formula">Bailey-Borwein-Plouffe formula</a>.

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

%F 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

%e 3.243f6a8885a308d3...

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

%o (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

%o (PARI) N=50; default(realprecision,.75*N); A062964=digits(Pi*16^N\1,16) \\ _M. F. Hasler_, Mar 14 2015

%Y 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).

%Y Cf. A007514, A374334.

%K easy,nonn,base,cons

%O 1,1

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

%E More terms from _Henry Bottomley_, Jul 24 2001