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A308879 a(n) is the n-digit integer m that maximizes sin(m). It's also the n-digit integer that minimizes the mean square error of the approximation sin(x+m) for cos(x) over [0, 2*Pi]. 0
8, 33, 699, 9929, 51819, 573204, 4846147, 37362253, 288632526, 9251925681, 81129397337, 881156436695 (list; graph; refs; listen; history; text; internal format)



Naturally, sin(a(n)) is the best approximation to 1 for an n-digit integer argument. a(n) is the closest integer to an n-digit 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.


Table of n, a(n) for n=1..12.


For n=3, a(3)=699 since no other 3-digit 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.



  double e = 1.0;

  int b = 0, d=1, c=10;

  int a[10]; // print A to see the results

  for( int i=0; d<10; i++ )


      double y = double(i*4+1)*PI/2.0;

      double z = round(y);

      double f = abs(z-y);

      int    w = int(z);

      if( w>=c ) { a[d]=b; c*=10; e=1.0; b=0; d++; }

      if( f< e ) { e=f; b=w; }



Cf. A046959.

Sequence in context: A069509 A204191 A041118 * A297904 A298498 A091720

Adjacent sequences:  A308876 A308877 A308878 * A308880 A308881 A308882




Inigo Quilez, Feb 12 2020



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Last modified January 22 10:48 EST 2021. Contains 340362 sequences. (Running on oeis4.)