A197733 -id:A197733 - cp's OEIS frontend

A197682 Decimal expansion of Pi/(2 + 2*Pi).

3, 7, 9, 2, 7, 3, 4, 9, 6, 4, 9, 7, 3, 8, 8, 0, 7, 2, 6, 7, 2, 2, 1, 5, 3, 4, 4, 5, 2, 2, 4, 4, 6, 4, 3, 2, 0, 6, 9, 2, 1, 3, 1, 8, 2, 8, 2, 0, 2, 6, 5, 4, 9, 8, 3, 3, 4, 4, 9, 4, 1, 0, 6, 8, 9, 1, 2, 7, 4, 0, 6, 8, 5, 5, 0, 4, 7, 8, 6, 8, 8, 1, 6, 0, 3, 1, 6, 5, 8, 7, 0, 0, 7, 6, 7, 7, 8, 8, 6

Offset: 0

Clark Kimberling, Oct 17 2011

The number Pi/(2 + 2*Pi) is the least x > 0 such that sin(x) = cos(Pi*x).
If b and c are distinct real numbers, the solutions of sin(bx) = cos(cx) are x = (k - 1/2)*Pi/(b + c), where k runs through the integers. Thus, if b > 0 and c > 0, the least solution x > 0 is Pi/(2*b + 2*c), so that this is also the least x > 0 for which sin(c*x) = cos(b*x).  Related sequences, each with a Mathematica program which includes a graph:
...
b.....c.......sequence........x
1.....2.......A019673........ x = Pi/6
1.....3.......A019678........ x = Pi/8
1.....4.......(A000796)/10... x = Pi/10
1.....Pi......A197682........ x = Pi/(2+2*Pi)
1.....2*Pi....A197683........ x = Pi/(2+4*Pi)
1.....1/Pi....A197684........ x = Pi^2/(2+2*Pi)
1.....2/Pi....A197685........ x = Pi^2/(4+2*Pi)
1.....Pi/2....A197686........ x = Pi/(2+Pi)
1.....Pi/3....A197687........ x = 3*Pi/(6+2*Pi)
1.....Pi/4....A197688........ x = 2*Pi/(4+Pi)
1.....Pi/6....A197689........ x = 3*Pi/(6+Pi)
2.....3.......(A000796)/10... x = Pi/10
2.....Pi......A197690........ x = Pi/(4+2*Pi)
2.....2*Pi....A197691........ x = Pi/(4+4*Pi)
2.....1/Pi....A197692........ x = Pi^2/(2+4*Pi)
2.....2/Pi....A197693........ x = Pi^2/(4+4*Pi)
2.....Pi/2....A197694........ x = Pi/(4+Pi)
3.....Pi......A197695........ x = Pi/(2+2*Pi)
3.....2*Pi....A197696........ x = Pi/(6+4*Pi)
3.....1/Pi....A197697........ x = Pi^2/(2+6*Pi)
3.....2/Pi....A197698........ x = Pi^2/(4+6*Pi)
3.....Pi/2....A197699........ x = Pi/(6+Pi)
1/2...Pi......A197700........ x = Pi/(1+2*Pi)
1/2...2*Pi....A197701........ x = Pi/(1+4*Pi)
1/2...1/Pi....A197724........ x = Pi^2/(2+Pi)
1/2...2/Pi....A197725........ x = Pi^2/(4+Pi)
1/2...Pi/2....A197726........ x = Pi/(1+Pi)
1/2...Pi/4....A197727........ x = 2*Pi/(2+Pi)
1/3...Pi/3....A197728........ x = 3*Pi/(2+2*Pi)
1/3...Pi/6....A197729........ x = 3*Pi/(2+Pi)
2/3...Pi/6....A197730........ x = 3*Pi/(4+Pi)
1/4...Pi......A197731........ x = 2*Pi/(1+4*Pi)
1/4...Pi/2....A197732........ x = 2*Pi/(1+2*Pi)
1/4...Pi/4....A197733........ x = 2*Pi/(1+Pi)
1/5...Pi/5....10*A197691..... x = 5*Pi/(2+2*Pi)
1/6...Pi/6....A197735........ x = 3*Pi/(1+Pi)
1/8...Pi/8....A197736........ x = 4*Pi/(1+Pi)

			0.37927349649738807267221534452244643...

Index entries for transcendental numbers

Cf. A197683.

b = 1; c = Pi;
t = x /. FindRoot[Sin[b*x] == Cos[c*x], {x, .3, .4}]
N[Pi/(2*b + 2*c), 110]
RealDigits[%]  (* A197682 *)
Simplify[Pi/(2*b + 2*c)]
Plot[{Sin[b*x], Cos[c*x]}, {x, 0, Pi}]

1/(2/Pi+2) \\ Charles R Greathouse IV, Sep 27 2022

A272873 Decimal expansion of the quadratic mean of 1 and Pi.

Original entry on oeis.org

2, 3, 3, 1, 2, 6, 6, 2, 2, 2, 5, 8, 0, 4, 8, 4, 1, 1, 6, 2, 1, 5, 2, 5, 3, 0, 1, 9, 4, 2, 9, 6, 8, 5, 7, 5, 1, 7, 3, 9, 6, 3, 3, 7, 7, 6, 9, 5, 5, 6, 6, 4, 4, 5, 9, 3, 0, 6, 8, 4, 0, 8, 8, 7, 3, 1, 8, 2, 5, 4, 6, 3, 7, 6, 1, 6, 7, 2, 3, 5, 8, 2, 2, 0, 8, 9, 5, 9, 0, 6, 9, 1, 7, 5, 4, 7, 7, 2, 2, 3, 5, 3, 7, 5, 5

Offset: 1

Stanislav Sykora, May 15 2016

Quadratic mean (also known as the root mean square, rms) of two numbers x and y, is the Hoelder mean H(x,y,p) = ((x^2+y^2)/2)^(1/p) with p = 2.

			2.3312662225804841162152530194296857517396337769556644593068408873...

Stanislav Sykora, Table of n, a(n) for n = 1..1000
Wikipedia, Generalized mean
Index entries for transcendental numbers

Cf. A000796, A002388.

Other means of 1 and Pi: A002161 (geometric, p=0), A191502 (AGM), A197733 (harmonic, p=-1), A269430 (arithmetic, p=1).

RealDigits[Sqrt[(1+Pi^2)/2],10,120][[1]] (* Harvey P. Dale, Apr 01 2018 *)

PARI
```
sqrt((1+Pi^2)/2)
```

Equals sqrt((1+Pi^2)/2).

cp's OEIS Frontend

A197682 Decimal expansion of Pi/(2 + 2*Pi).

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