| Displaying 1-9 of 9 results found.
page
1
0, 3, 6, 9, 12, 15, 18, 21, 25, 28, 31, 34, 37, 40, 43, 47, 50, 53, 56, 59, 62, 65, 69, 72, 75, 78, 81, 84, 87, 91, 94, 97, 100, 103, 106, 109, 113, 116, 119, 122, 125, 128, 131, 135, 138, 141, 144, 147, 150, 153, 157, 160, 163, 166, 169, 172, 175, 179, 182, 185, 188, 191, 194
COMMENTS
Beatty sequence for Pi.
These are the nonnegative integers m satisfying sin(m)*sin(m+1) <= 0. In general, the Beatty sequence of an irrational number r > 1 consists of the numbers m satisfying sin(m*x)*sin((m+1)*x) <= 0, where x = Pi/r. Thus the numbers m satisfying sin(m*x)*sin((m+1)*x) > 0 form the Beatty sequence of r/(1-r). - Clark Kimberling, Aug 21 2014
This can also be stated in terms of the tangent function. These are the nonnegative integers m such that tan(m/2)*tan(m/2+1/2) <= 0. In general, the Beatty sequence of an irrational number r > 1 consists of the numbers m satisfying tan(m*x/2)*tan((m+1)*x/2) <= 0, where x = Pi/r. Thus the numbers m satisfying tan(m*x/2)*tan((m+1)*x/2) > 0 form the Beatty sequence of r/(1-r). - Clark Kimberling, Aug 22 2014
FORMULA
a(n)/n converges to Pi because |a(n)/n - Pi| = |a(n) - n*Pi|/n < 1/n. - Hieronymus Fischer, Jan 22 2006
EXAMPLE
a(7)=21 because 7*Pi=21.9911... and a(8)=25 because 8*Pi=25.1327.... a(100000)=314159 because Pi=3.141592...
PROG
(PARI) vector(80, n, n--; floor(n*Pi)) \\ G. C. Greubel, Sep 28 2018 (Magma) R:=RieldField(10); [Floor(n*Pi(R)): n in [0..80]]; // G. C. Greubel, Sep 28 2018
Self-inverse integer permutation induced by Beatty sequences for Pi and Pi/(Pi-1).
+10
7
3, 6, 1, 9, 12, 2, 15, 18, 4, 21, 25, 5, 28, 31, 7, 34, 37, 8, 40, 43, 10, 47, 50, 53, 11, 56, 59, 13, 62, 65, 14, 69, 72, 16, 75, 78, 17, 81, 84, 19, 87, 91, 20, 94, 97, 100, 22, 103, 106, 23, 109, 113, 24, 116, 119, 26, 122, 125, 27, 128, 131, 29, 135, 138, 30, 141, 144
1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 82, 83, 84, 85
COMMENTS
Beatty sequence for 2*Pi/(2*Pi-1); complement of A038130; not the same as A108120: a(37)=44 <> A108120(37)=43.
MATHEMATICA
With[{c=2Pi}, Floor[(c*Range[80])/(c-1)]] (* Harvey P. Dale, Apr 21 2024 *)
a(n) = floor(n*Pi/(Pi-2)).
+10
5
2, 5, 8, 11, 13, 16, 19, 22, 24, 27, 30, 33, 35, 38, 41, 44, 46, 49, 52, 55, 57, 60, 63, 66, 68, 71, 74, 77, 79, 82, 85, 88, 90, 93, 96, 99, 101, 104, 107, 110, 112, 115, 118, 121, 123, 126, 129, 132, 134, 137, 140, 143, 145, 148, 151, 154, 156, 159, 162, 165, 167
COMMENTS
Beatty sequence for Pi/(Pi-2); complement of A140758.
MATHEMATICA
With[{c=Pi/(Pi-2)}, Floor[c*Range[70]]] (* Harvey P. Dale, Apr 19 2014 *)
PROG
(Magma) R:= RealField(40); [Floor(n*Pi(R)/(Pi(R)-2)): n in [1..60]]; // G. C. Greubel, Oct 21 2023 (SageMath) [floor(n*pi/(pi-2)) for n in range(1, 61)] # G. C. Greubel, Oct 21 2023
Nonnegative integers k satisfying sin(k) >= 0 and sin(k+1) <= 0.
+10
5
3, 9, 15, 21, 28, 34, 40, 47, 53, 59, 65, 72, 78, 84, 91, 97, 103, 109, 116, 122, 128, 135, 141, 147, 153, 160, 166, 172, 179, 185, 191, 197, 204, 210, 216, 223, 229, 235, 241, 248, 254, 260, 267, 273, 279, 285, 292, 298, 304, 311, 317, 323, 329, 336, 342
MATHEMATICA
z = 400; f[x_] := Sin[x]
Select[Range[0, z], f[#]*f[# + 1] <= 0 &] (* A022844 *) Select[Range[0, z], f[#] >= 0 && f[# + 1] <= 0 &] (* A246388 *) Select[Range[0, z], f[#] <= 0 && f[# + 1] >= 0 &] (* A038130 *) Select[Range[0, z], f[#]*f[# + 1] > 0 &] (* A054386 *) Select[Range[0, z], f[#] >= 0 && f[# + 1] >= 0 &] (* A246389 *) Select[Range[0, z], f[#] <= 0 && f[# + 1] <= 0 &] (* A246390 *)
a(1) = floor(Pi) = 3; a(n+1) = floor(a(n)*Pi).
+10
4
3, 9, 28, 87, 273, 857, 2692, 8457, 26568, 83465, 262213, 823766, 2587937, 8130243, 25541911, 80242279, 252088554, 791959549, 2488014301, 7816327450, 24555716894, 77144059797, 242355211526, 761381352089, 2391950062303
COMMENTS
a(n+1)/a(n) converges to Pi. Similar to sequence A085839 but with a simpler definition. Subset of the Beatty sequence of Pi = A022844 = floor(n*Pi). Primes in this sequence include a(1) = 3, a(6) = 857, a(15) = 25541911. - Jonathan Vos Post, Jan 18 2006
EXAMPLE
a(2) = floor(a(1)*Pi) = floor(3*Pi) = 9;
a(3) = floor(a(2)*Pi) = floor(9*Pi) = 28;
a(4) = floor(a(3)*Pi) = floor(28*Pi) = 87.
MAPLE
A[1]:= 3:
for n from 2 to 50 do A[n]:= floor(Pi*A[n-1]) od:
MATHEMATICA
a[1] = Floor[Pi]; a[n_] := a[n] = Floor[a[n - 1]*Pi]; Array[a, 25] (* Robert G. Wilson v, Jan 18 2006 *)
Nonnegative integers k satisfying sin(k) >= 0 and sin(k+1) >= 0.
+10
4
0, 1, 2, 7, 8, 13, 14, 19, 20, 26, 27, 32, 33, 38, 39, 44, 45, 46, 51, 52, 57, 58, 63, 64, 70, 71, 76, 77, 82, 83, 88, 89, 90, 95, 96, 101, 102, 107, 108, 114, 115, 120, 121, 126, 127, 132, 133, 134, 139, 140, 145, 146, 151, 152, 158, 159, 164, 165, 170, 171
MAPLE
Digits := 100:
isA246389 := proc(k)
if evalf(sin(k)) >= 0 and evalf(sin(k+1)) >= 0 then
return true ;
else
return false ;
end if;
end proc:
option remember ;
if n = 1 then
0;
else
for a from procname(n-1)+1 do
if isA246389(a) then
return a;
end if;
end do:
end if;
end proc:
MATHEMATICA
z = 400; f[x_] := Sin[x]
Select[Range[0, z], f[#]*f[# + 1] <= 0 &] (* A022844 *) Select[Range[0, z], f[#] >= 0 && f[# + 1] <= 0 &] (* A246388 *) Select[Range[0, z], f[#] <= 0 && f[# + 1] >= 0 &] (* A038130 *) Select[Range[0, z], f[#]*f[# + 1] > 0 &] (* A054386 *) Select[Range[0, z], f[#] >= 0 && f[# + 1] >= 0 &] (* A246389 *) Select[Range[0, z], f[#] <= 0 && f[# + 1] <= 0 &] (* A246390 *)
Beatty sequence for 1/(e^Pi - Pi^e), complement of A127451.
+10
3
1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 24, 26, 27, 29, 30, 32, 33, 35, 36, 38, 39, 41, 42, 44, 45, 46, 48, 49, 51, 52, 54, 55, 57, 58, 60, 61, 63, 64, 66, 67, 68, 70, 71, 73, 74, 76, 77, 79, 80, 82, 83, 85, 86, 88, 89, 90, 92, 93, 95, 96, 98, 99, 101, 102
FORMULA
a(n) = floor(n/(e^Pi - Pi^e))
MATHEMATICA
Table[Floor[n/(Exp[Pi] - Pi^E)], {n, 70}]
Nonnegative integers k satisfying sin(k) <= 0 and sin(k+1) <= 0.
+10
3
4, 5, 10, 11, 16, 17, 22, 23, 24, 29, 30, 35, 36, 41, 42, 48, 49, 54, 55, 60, 61, 66, 67, 68, 73, 74, 79, 80, 85, 86, 92, 93, 98, 99, 104, 105, 110, 111, 112, 117, 118, 123, 124, 129, 130, 136, 137, 142, 143, 148, 149, 154, 155, 156, 161, 162, 167, 168, 173
MATHEMATICA
z = 400; f[x_] := Sin[x]
Select[Range[0, z], f[#]*f[# + 1] <= 0 &] (* A022844 *) Select[Range[0, z], f[#] >= 0 && f[# + 1] <= 0 &] (* A246388 *) Select[Range[0, z], f[#] <= 0 && f[# + 1] >= 0 &] (* A038130 *) Select[Range[0, z], f[#]*f[# + 1] > 0 &] (* A054386 *) Select[Range[0, z], f[#] >= 0 && f[# + 1] >= 0 &] (* A246389 *) Select[Range[0, z], f[#] <= 0 && f[# + 1] <= 0 &] (* A246390 *) SequencePosition[Table[If[Sin[n]<=0, 1, 0], {n, 200}], {1, 1}][[;; , 1]] (* Harvey P. Dale, Apr 02 2023 *)
Search completed in 0.019 seconds
| 