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Sin(n) decreases monotonically to -1.
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5
1, 3, 4, 5, 11, 344, 1054, 1764, 2474, 3184, 3894, 4604, 5314, 6024, 6734, 7444, 8154, 8864, 9574, 10284, 10994, 11704, 12414, 13124, 13834, 14544, 15254, 15964, 16674, 17384, 18094, 18804, 19514, 20224, 20934, 21644, 22354, 23064, 23774, 24484, 25194, 25904
COMMENTS
Sin(10265498) =-0.9999999999999999313932793053103935998520142594607...
MATHEMATICA
z={}; current=1; Timing[ Do[ If[ Sin[ n ]<current, AppendTo[ z, current=Sin[ n ] ] ], {n, 30000} ] ]; z
d = 1; lst = {}; Do[a = Sin@n; If[a > d, d = a; Print@n; AppendTo[lst, n]], {n, 111111111}]; lst (* Robert G. Wilson v, Aug 24 2007 *)
PROG
(PARI) d=oo; print1("1, 3, "); for(k=1, 10^8, my(di=2*k/Pi, dir=round(di), dd); if(dir%4==3, dd=abs(di-dir); if(dd<d, print1(k, ", "); d=dd))) \\ Hugo Pfoertner, Feb 29 2020
Permutation of the positive integers related to the "Sine tree" (see Comments lines for construction details).
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2
1, 3, 2, 4, 8, 9, 5, 7, 10, 18, 16, 37, 21, 15, 43, 75, 33, 36, 20, 14, 11, 19, 17, 34, 38, 22, 29, 41, 73, 66, 150, 86, 31, 172, 300, 132, 147, 83, 59, 44, 76, 68, 35, 39, 6, 13, 79, 71, 136, 152, 88, 119, 167, 295, 264, 600, 344, 62, 173, 301, 133, 146, 82
COMMENTS
The "Sine tree" is constructed according to this procedure:
take an infinite complete binary tree,
initially, the nodes have no value,
for each n=1,2,3,...:
move to the root node,
while the current node has a value:
if sin(n)<sin(node.value) then move to the left child node,
else move to the right child node.
assign the value n to the current node.
As the set {sin(1), sin(2), sin(3), ...} is dense in the open interval ]-1, +1[, each node will eventually have a value.
a(n) corresponds to the "index" of the node with value n:
- the index of the root node is 1,
- the index of the left child of the node with index k is 2*k,
- the index of the right child of the node with index k is 2*k+1.
a( A046959(n)) = 2^(n-1)-1, for any n>1. a( A046964(n)) = 2^(n-1), for any n>0.
EXAMPLE
For n=1: the root node has no value, so we assign it the value 1, and a(1)=1.
For n=2: the root node has value 1, and sin(2)>sin(1), so we move to the right child node. This node has no value, so we assign it the value 2, and a(2)=2*1+1.
For n=3: the root node has value 1, and sin(3)<sin(1), so we move to the left child node. This node has no value, so we assign it the value 3, and a(3)=2*1.
PROG
(Perl) See Links section.
1, 3, 2, 4, 7, 45, 8, 5, 6, 9, 21, 310, 46, 20, 14, 11, 23, 10, 22, 19, 13, 26, 334, 378, 104038, 89, 309, 335, 27, 341, 33, 344, 17, 24, 43, 18, 12, 25, 44, 336
COMMENTS
a(2^n-1)= A046959(n+1), for any n>0.
a(1) = 1, and for each n >= 2, a(n) is the smallest number such that 1/sin(a(n)) < 1/sin(a(k)) for all k < n, so that 1/sin(a(1)) > 1/sin(a(2)) > ... > 1/sin(a(n)) > ...
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0
1, 2, 4, 6, 22, 333, 355, 103993, 104348, 1042060, 1146408, 4272943, 5419351, 80143857
REFERENCES
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 83, p. 29, Ellipses, Paris 2008. Also Entry 137, p. 47.
LINKS
Eric Weisstein's World of Mathematics, Pi
EXAMPLE
1/sin(1) = 1.1883951; 1/sin(2) = 1.0997501; 1/sin(4) = - 1.3213487.
MAPLE
a:= evalf(1/sin(1)); for n from 2 to 10000000 do; if a > evalf(1/sin(n)) then a:= evalf(1/sin(n)); print(n); else fi ; od;
MATHEMATICA
vm = 2; s = {}; Do[v = 1/Sin[n]; If[v < vm, vm = v; AppendTo[s, n]], {n, 1, 110000}]; s (* Amiram Eldar, Aug 10 2019 *)
PROG
(PARI) lista(NN) = {my(x=2); for(k=1, NN, if(1/sin(k)<x, x=1/sin(k); print1(k", "))); } \\ Jinyuan Wang, Aug 12 2019
EXTENSIONS
a(13) corrected and a(14) added by Amiram Eldar, Aug 10 2019
a(n) is the n-digit integer m that maximizes sin(m).
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0
8, 33, 699, 9929, 51819, 573204, 4846147, 37362253, 288632526, 9251925681, 81129397337, 881156436695
COMMENTS
a(n) is also the n-digit integer that minimizes the mean square error of the approximation sin(x+m) for cos(x) over [0, 2*Pi].
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.
EXAMPLE
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.
PROG
(C)
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; }
}
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