A195284 -id:A195284 - cp's OEIS frontend

A197155 Decimal expansion of the shortest distance from the x axis through (4,1) to the line y=x/2.

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

Offset: 1

Clark Kimberling, Oct 11 2011

The shortest segment from one side of an angle T through a point P inside T is called the Philo line of P in T. For discussions and guides to related sequences, see A197032, A197008 and A195284.

			length of Philo line:  1.94634640...
endpoint on x axis:    (4.2236, 0); see A197154
endpoint on line y=3x: (3.79888, 1.89944)

Cf. A197032, A197155, A197008, A195284.

f[t_] := (t - k*t/(k + m*t - m*h))^2 + (m*k*t/(k + m*t - m*h))^2;
g[t_] := D[f[t], t]; Factor[g[t]]
p[t_] :=  h^2 k + k^3 - h^3 m - h k^2 m - 3 h k t + 3 h^2 m t + 2 k t^2 - 3 h m t^2 + m t^3
m = 1/2; h = 4; k = 1;(* slop m, point (h,k) *)
t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]
RealDigits[t]  (* A197154 *)
{N[t], 0} (* endpoint on x axis *)
{N[k*t/(k + m*t - m*h)],
 N[m*k*t/(k + m*t - m*h)]} (* endpt on line y=x/2 *)
d = N[Sqrt[f[t]], 100]
RealDigits[d]  (* A197155 *)
Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 4.5}],
 ContourPlot[(x - h)^2 + (y - k)^2 == .002, {x, 0, 4.5}, {y, 0, 3}],
 PlotRange -> {0, 2}, AspectRatio -> Automatic]

A195285 Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of a 3,4,5 right triangle ABC.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Sep 14 2011

See A195284 for a definition of Philo(ABC,I) and general discussion.

			Philo(ABC,I)=0.59772335075207494572...

Cf. A002163, A010466, A195284.

a = 3; b = 4; c = 5;
h = a (a + c)/(a + b + c); k = a*b/(a + b + c);
f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2;
s = NSolve[D[f[t], t] == 0, t, 150]
f1 = (f[t])^(1/2) /. Part[s, 4]
RealDigits[%, 10, 100] (* (A) A195284 *)
f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
s = NSolve[D[f[t], t] == 0, t, 150]
f3 = (f[t])^(1/2) /. Part[s, 1]
RealDigits[%, 10, 100] (* (B) A002163 *)
f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
s = NSolve[D[f[t], t] == 0, t, 150]
f2 = (f[t])^(1/2) /. Part[s, 4]
RealDigits[%, 10, 100] (* (C) A010466 *)
(f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,I) A195285 *)

A197033 Decimal expansion of the shortest distance from the x axis through (2,1) to the line y=x.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Oct 10 2011

For discussions and guides to related sequences, see A197032, A197008 and A195284.

			length of Philo line:  1.8442716817001718647799577442735702984134...
endpoint on x axis:  (2.35321..., 0); see A197032
endpoint on line y=x:  (1.73898, 1.73898)

Cf. A197032, A197008, A195284.

f[t_] := (t - k*t/(k + m*t - m*h))^2 + (m*k*t/(k + m*t - m*h))^2;
g[t_] := D[f[t], t]; Factor[g[t]]
p[t_] := h^2 k + k^3 - h^3 m - h k^2 m - 3 h k t + 3 h^2 m t + 2 k t^2 - 3 h m t^2 + m t^3 (* root of p[t] minimizes f *)
m = 1; h = 2; k = 1; (* m=slope; (h,k)=point *)
t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]
RealDigits[t]  (* A197032 *)
{N[t], 0} (* lower endpoint of minimal segment [Philo line] *)
{N[k*t/(k + m*t - m*h)],
N[m*k*t/(k + m*t - m*h)]} (* upper endpoint *)
d = N[Sqrt[f[t]], 100]
RealDigits[d] (* A197033 *)
Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 2.5}],
 ContourPlot[(x - h)^2 + (y - k)^2 == .003, {x, 0, 3}, {y, 0, 3}], PlotRange -> {0, 2}, AspectRatio -> Automatic]

A137262 Floor of sum of the first 10^n square roots.

Original entry on oeis.org

1, 22, 671, 21097, 666716, 21082008, 666667166, 21081852648, 666666671666, 21081851083600, 666666666716666, 21081851067947309, 666666666667166666, 21081851067790776685, 666666666666671666666, 21081851067789211358047, 666666666666666716666666, 21081851067789195704773173

Offset: 0

Cino Hilliard, Mar 12 2008

nonn

			The first 10^0 square roots is 1. The sum of the first 10^1 square roots is 22.468278186... . So 22 is the second entry in the sequence.

Amiram Eldar, Table of n, a(n) for n = 0..666

Cf. A025224, A195284.

a[n_] := Floor[Sum[Sqrt[j],{j,10^n}]];Array[a,17,0] (* James C. McMahon, May 17 2025 *)
a[n_] := Floor[HarmonicNumber[10^n, -1/2]]; Array[a, 20, 0] (* Amiram Eldar, Jul 18 2025 *)

g2(n,p=2) = for(j=0,n,s=0;for(x=0,10^j,s+=x^(1/p)); print1(floor(s)", "))

From Amiram Eldar, Jul 18 2025: (Start)
a(n) = A025224(10^n).
a(n) ~ (2/3) * 10^(3*n/2). (End)

a(10)-a(16) from James C. McMahon, May 17 2025

a(17) from Amiram Eldar, Jul 18 2025

cp's OEIS Frontend

A197155 Decimal expansion of the shortest distance from the x axis through (4,1) to the line y=x/2.

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A195285 Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of a 3,4,5 right triangle ABC.

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A197033 Decimal expansion of the shortest distance from the x axis through (2,1) to the line y=x.

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Mathematica

A137262 Floor of sum of the first 10^n square roots.

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