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Pi = Sum_{n >= 0} a(n)/n!.
(Formerly M2193)
32

%I M2193 #25 Mar 22 2017 21:56:09

%S 3,0,0,0,3,1,5,6,5,0,1,4,7,8,0,6,7,10,7,10,4,10,6,16,1,11,20,3,18,12,

%T 9,13,18,21,14,34,27,11,27,33,36,18,5,18,5,23,39,1,10,42,28,17,20,51,

%U 8,42,47,0,27,23,16,52,32,52,53,24,43,61,64,18,17,11,0,53,14,62

%N Pi = Sum_{n >= 0} a(n)/n!.

%C The current name does not define a(n) without ambiguity. It is meant that for each n, a(n) is the largest integer such that the remainder of Pi - (partial sum up to n) remains positive. This leads to the FORMULA given below. - _M. F. Hasler_, Mar 20 2017

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Hans Havermann, <a href="/A007514/b007514.txt">Table of n, a(n) for n = 0..10000</a>

%H <a href="/index/Ph#Pi314">Index entries for sequences related to the number Pi</a>

%F a(n) = floor(n!*Pi) - n*floor((n-1)!*Pi) for all n > 0. - _M. F. Hasler_, Mar 20 2017

%e Pi = 3/0! + 0/1! + 0/2! + 0/3! + 3/4! + 1/5! + ...

%t p = N[Pi, 1000]; Do[k = Floor[p*n! ]; p = p - k/n!; Print[k], {n, 0, 75} ]

%o (PARI) x=Pi;vector(floor((y->y/log(y))(default(realprecision))),n,t=(n-1)!;k=floor(x*t);x-=k/t;k) \\ _Charles R Greathouse IV_, Jul 15 2011

%o (PARI) C=1/Pi;x=0;vector(primepi(default(realprecision)),n,-x*n--+x=n!\C) \\ _M. F. Hasler_, Mar 20 2017

%Y Essentially same as A075874.

%Y Pi in base n: A004601 to A004608, A000796, A068436 to A068440, A062964.

%K nonn

%O 0,1

%A _N. J. A. Sloane_, _Robert G. Wilson v_