%I
%S 8,33,699,9929,51819,573204,4846147,37362253,288632526,9251925681,
%T 81129397337,881156436695
%N a(n) is the ndigit integer m that maximizes sin(m). It's also the ndigit integer that minimizes the mean square error of the approximation sin(x+m) for cos(x) over [0, 2*Pi].
%C Naturally, sin(a(n)) is the best approximation to 1 for an ndigit integer argument. a(n) is the closest integer to an ndigit number of the form (4k+1)*Pi/2. Often used to compute an approximated rotation matrix with just a few number of characters of code, as in M = sin(x+{0,699,699,0}). It is not guaranteed that each term in the sequence produces a better approximation than the previous one, although numerical evidence suggests so. It is therefore also not guaranteed to be a subsequence of A046959.
%e For n=3, a(3)=699 since no other 3digit integer m makes sin(x+m) closer to cos(x) than m=699 does. For example cos(4.5)=0.210795799... and sin(4.5+699)=0.215061112... and no other value of m will make the latter closer to the former.
%o (C)
%o double e = 1.0;
%o int b = 0, d=1, c=10;
%o int a[10]; // print A to see the results
%o for( int i=0; d<10; i++ )
%o {
%o double y = double(i*4+1)*PI/2.0;
%o double z = round(y);
%o double f = abs(zy);
%o int w = int(z);
%o if( w>=c ) { a[d]=b; c*=10; e=1.0; b=0; d++; }
%o if( f< e ) { e=f; b=w; }
%o }
%Y Cf. A046959.
%K nonn,base,more
%O 1,1
%A _Inigo Quilez_, Feb 12 2020
