A331313 - OEIS

%I #38 Jan 06 2025 21:52:18

%S 1,0,0,1,1,-1,1,1,1,-1,0,0,0,-1,0,1,1,-1,1,1,0,1,-1,1,1,1,1,1,1,0,0,

%T -1,0,0,0,0,-1,1,-1,-1,-1,-1,1,-1,0,-1,0,-1,0,-1,-1,0,1,-1,-1,1,1,0,0,

%U 1,0,0,1,1,-1,0,0,-1,1,0,0,0,-1,-1,1,-1,-1,1,-1

%N Expansion of Pi in balanced ternary.

%C The correspondence to A004602 (see Formula section) can be seen by comparing the leading terms of each sequence from right to left and adding a carry when A004602 is 2:

%C A004602:  1, 0, 0, 1, 0, 2, 1, 1, 0, 1, 2, 2, 2, 2, 0, 1, 0, 2, 1, 1, 0, 0, 2, 1

%C   a(n) :  1, 0, 0, 1, 1,-1, 1, 1, 1,-1, 0, 0, 0,-1, 0, 1, 1,-1, 1, 1, 0, 1,-1, 1

%H Iain Fox, <a href="/A331313/b331313.txt">Table of n, a(n) for n = 2..20000</a>

%H Thomas König, <a href="/A331313/a331313.c.txt">Program to calculate the sequence with C and the GNU mpfr library</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Balanced_ternary">Balanced ternary</a>

%F Calculation can be done from A004602:

%F Choose n so that A004602(n) does not equal 2

%F Initialize {f(n)} from A004602 up to index n

%F Let i loop from n down to 2

%F   while f(i) is larger than 1

%F     set f(i) to f(i) - 3

%F     set f(i - 1) to f(i - 1) + 1

%F Set {a(n)} to {f(n)}

%e 10.011T111T000T011T1101T11111100T0000T...

%o (C) /* See König link. */

%o (PARI) \\ (adjust realprecision as needed)

%o first(n) = {default(realprecision, 10000); for(x=-1, +oo, v=digits(floor(Pi*3^(n+x)), 3); if(v[#v]!=1, break())); while(vecmax(v)==2, for(x=1, #v, if(v[x]==2, v[x]=-1; v[x-1]++))); vecextract(v,2^n-1)} \\ _Iain Fox_, Feb 03 2020

%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), A062964 (b=16), A060707 (b=60).

%Y Expansion of e in balanced ternary: A331990.

%K sign,base,easy

%O 2

%A _Thomas König_, Jan 13 2020