A197686 -id:A197686 - 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

A354222 Decimal expansion of 2 / (Pi+2).

Original entry on oeis.org

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

Offset: 0

Bernard Schott, May 19 2022

The sequence u(n+1) = 1 / ( u(n) + 1/(n+1) ) with u(0) >= 0 is convergent with lim_{n->oo} u(n) = 1; however, this sequence is monotonic iff u(0) = 2 / (Pi+2), and in this case, this sequence is increasing.

			0.388984529648342710619404612046031138262577367...

J. M. Arnaudiès, P. Delezoide et H. Fraysse, Exercices résolus d'Analyse du cours de mathématiques - 2, Dunod, 1993, Exercice 1, pp. 29-35.

Index entries for transcendental numbers

Cf. A060294, A197686.

Maple
```
evalf(2/(Pi+2), 100);
```

RealDigits[2/(Pi + 2), 10, 100][[1]] (* Amiram Eldar, May 19 2022 *)

PARI
```
2/(Pi+2) \\ Michel Marcus, May 20 2022
```

Equals 1 - A197686.

cp's OEIS Frontend

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

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