A019670 -id:A019670 - cp's OEIS frontend

A204067 Decimal expansion of the Fresnel Integral, Integral_{x >= 0} cos(x^3) dx.

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

Offset: 0

R. J. Mathar, Jan 10 2013

			0.7733429420779898501961016...

Muniru A Asiru, Table of n, a(n) for n = 0..2000
R. J. Mathar, Series expansion of generalized Fresnel integrals, arXiv:1211.3963 [math.CA], 2012, eq. (3.8).
Wikipedia, Fresnel Integral.

Cf. A010469, A019670, A073005, A073006, A204068, A217481.

evalf(int(cos(x^3),x=0..infinity),120); # Muniru A Asiru, Sep 26 2018

RealDigits[Gamma[1/3]/(2*Sqrt[3]), 10, 120][[1]] (* Amiram Eldar, May 26 2023 *)

Pi/(3*gamma(2/3)) \\ Gheorghe Coserea, Sep 26 2018

intnum(x=[0, -2/3], [oo, I], cos(x)/x^(2/3))/3 \\ Gheorghe Coserea, Sep 26 2018

Equals Pi/(3*Gamma(2/3)) = A019670 / A073006.

Equals Gamma(1/3)/(2*sqrt(3)) = A073005 / A010469. - Amiram Eldar, May 26 2023

A238238 Decimal expansion of the polar angle, in radians, of a cone which makes a golden-ratio cut of the full solid angle.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Feb 20 2014

The polar angle (or apex angle) of a cone which cuts a fraction f of the full solid angle (i.e., subtends a solid angle of 4*Pi*f steradians) is given by arccos(1-2*f). For a golden cut of the sphere surface by a cone with apex in its center, set f = 1-1/phi, phi being the golden ratio A001622. This value is in radians, its equivalent in degrees is A238239.

The apex angle of the isosceles triangle of smallest perimeter which circumscribes a semicircle (DeTemple, 1992). - Amiram Eldar, Jan 22 2022

			1.3324788649850305102080097919555854413349802774518956856629476856...

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Duane W. DeTemple, The Triangle of Smallest Perimeter which Circumscribes a Semicircle, The Fibonacci Quarterly, Vol. 30, No. 3 (1992), p. 274.
Wikipedia, Solid angle.

Cf. A001622, A019670, A137914, A238239.

RealDigits[ArcCos[2/GoldenRatio  -1],10,120][[1]] (* Harvey P. Dale, Jul 05 2019 *)

PARI
```
acos(4/(1+sqrt(5))-1)
```

arccos(1-2*(1-1/phi)) = arccos(2/phi-1), with phi = A001622.

A352453 Decimal expansion of the area of intersection of 4 unit-radius circles that have the vertices of a unit-side square as centers.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Mar 16 2022

The solution to a problem in Jones (1932): "At each corner of a garden, surrounded by a wall n yards square, a goat is tied with a rope n yards long. Find the area of the part of the garden common to the four goats." (When the square is taken to be of unit size, the common area is this constant.)
The perimeter of the shape formed by the intersection is 2*Pi/3 (A019693).
The solution to the three-dimensional version of this problem is A352454.

			0.31514674362772045262676811958729526112291787931465...

Donald L. Chambers, Problem 3684, School Science and Mathematics, Vol. 77, No. 5 (1977), p. 443; Solution by J. Philip Smith, ibid., Vol. 78, No. 4 (1978), pp. 354-355.
Amiram Eldar, Illustration.
Samuel Isaac Jones, Mathematical Nuts: For Lovers of Mathematics, 1932, Problems 9 and 10, pp. 86, 301-302.
Missouri State University, Problem #8, Finding the Area (resp. Volume) of Overlapping Circles (resp. Spheres), Advanced Problem Archive,; Solution to Problem #8, by Raymond Roan.
Bruce Shawyer, Problem 6, APICS 1999 Mathematics Competition, The Academy Corner, Crux Mathematicorum, Vol. 25, No. 8, 1999, p. 453; Solutions by Richard Tod and Catherine Shevlin, Vol. 26, No. 4, 2000, pp. 193-194.
Charles W. Trigg, Problem 686, Crux Mathematicorum, Vol. 7, No. 9, 1981, p. 275; Solution by Jordan Dou, Vol. 8, No. 9, 1982, p. 294.

Cf. A002194, A019670, A019693.

Cf. A075838, A133731, A192930, A352454.

RealDigits[1 + Pi/3 - Sqrt[3], 10, 100][[1]]

Equals 1 + Pi/3 - sqrt(3) = 1 + A019670 - A002194.

A281452 Expansion of f(x, x) * f(x^5, x^13) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

1, 2, 0, 0, 2, 1, 2, 0, 0, 4, 0, 0, 0, 1, 4, 0, 2, 2, 0, 0, 0, 2, 2, 0, 0, 2, 0, 0, 1, 4, 2, 0, 2, 0, 0, 0, 2, 2, 2, 0, 0, 2, 0, 0, 3, 2, 0, 0, 2, 4, 0, 0, 0, 4, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0, 4, 0, 0, 0, 0, 5, 2, 0, 0, 2, 0, 0, 0, 4, 2, 0, 2, 2, 0, 0, 0, 2, 2

Offset: 0

Michael Somos, Jan 26 2017

nonn

			G.f. = 1 + 2*x + 2*x^4 + x^5 + 2*x^6 + 4*x^9 + x^13 + 4*x^14 + 2*x^16 + ...
G.f. = q^4 + 2*q^13 + 2*q^40 + q^49 + 2*q^58 + 4*q^85 + q^121 + 4*q^130 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A002654, A004018, A019670, A035154, A105673, A113446, A122856, A122857, A122864, A122865.
Cf. A125061, A129448, A138949, A138950, A163746, A256269, A256276, A256280, A258034.
Cf. A258228, A258256, A258278, A258292, A259761.
Cf. A281451, A281453, A281490, A281491, A281492.

a[ n_] := If[ n < 0, 0, DivisorSum[ 9 n + 4, KroneckerSymbol[ -4, #] &]];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] QPochhammer[ -x^5, x^18] QPochhammer[ -x^13, x^18] QPochhammer[ x^18], {x, 0, n}];
a[ n_] := If[ n < 0, 0, Times @@ (Which[ # < 3, 1, Mod[#, 4] == 1, #2 + 1, True, (1 + (-1)^#2) / 2] & @@@ FactorInteger[ 9 n + 4])];

{a(n) = if( n<0, 0, sumdiv(9*n + 4, d, (d%4==1) - (d%4==3)))};

{a(n) = if( n<0, 0, my(m = 9*n + 4, k, s); forstep(j=0, sqrtint(m), 3, if( issquare(m - j^2, &k) && (k%9 == 2 || k%9 == 7), s+=(j>0)+1)); s)};

{a(n) = if( n<0, 0, my(A, p, e); A = factor(9*n + 4); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if(p==2, 1, p%4==1, e+1, 1-e%2)))};

f(a,b) = 1 + Sum_{k=1..oo} (ab)^(k(k-1)/2)*(a^k+b^k). - N. J. A. Sloane, Jan 30 2017
Euler transform of a period 36 sequence.
G.f.: (Sum_{k in Z} x^k^2) * (Sum_{k in Z} x^(9*k^2 + 4*k)).
G.f.: Product_{k>0} (1 + x^(2*k-1))^2 * (1 - x^(2*k)) * (1 - x^(18*k-13)) * (1 - x^(18*k-5)) * (1 - x^(18*k)).
a(n) = A122865(3*n + 1) = A122856(6*n + 2) = A258278(6*n + 2). a(n) = - A256269(9^n + 4). 4 * a(n) = A004018(9*n + 4).
a(n) = b(9*n + 4) with b = A002654, A035154, A113446, A122864, A125061, A129448, A138950, A163746, A256276, A258228, A258256.
2 * a(n) = b(9*n + 4) = with b = A105673, A105673, A122857, A258034, A259761. -2 * a(n) = b(9*n + 4) with b = A138949, A256280, A258292.
a(4*n) = A281453(n). a(8*n + 6) = 2 * A281490(n). a(16*n + 12) = A281451(n).
a(32*n + 4) = 2 * A281492(n). a(64*n + 28) = A281452(n). a(128*n + 60) = 2 * A281491(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/3 = 1.0471975... (A019670). - Amiram Eldar, Jan 20 2025

A346573 Decimal expansion of 2 - Pi/3.

Original entry on oeis.org

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

Offset: 0

Sean A. Irvine, Jul 23 2021

			0.9528024488034022538457855389...

L. B. W. Jolley, Summation of Series, Dover, 1961, Eq. (316).

Michael I. Shamos, Shamos's catalog of the real numbers (2011).
Index entries for transcendental numbers.

Cf. A019670, A346572, A346571, A346570.

RealDigits[2-Pi/3, 10, 120][[1]] (* Amiram Eldar, Jun 13 2023 *)

Equals 2 - A019670.

Equals 1 + Sum_{k>=1} ( ((-1)^k/(6*k-1) - (-1)^k/(6*k+1)) ).

A152627 Decimal expansion of 3/4.

Original entry on oeis.org

7, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane, Oct 30 2009

This is also the expected distance from a randomly selected point in the unit sphere to its center. - Derek Orr, Jul 27 2014

			0.75000000000000000000000000000000000000000000000000000000000...

Cf. A019670 (Pi/3).

RealDigits[3/4, 10, 128][[1]] (* Alonso del Arte, Feb 16 2015 *)

Equals (sin(Pi/3))^2. - Michel Marcus, Sep 12 2020

Equals Sum_{k>=1} (zeta(2*k) - 1). - Amiram Eldar, May 24 2021

A274399 a(n) = floor(n*Pi/3).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70

Offset: 0

Paulo Romero Zanconato Pinto, Sep 26 2016

			For n = 1000 we have that floor(1000*Pi/3) = floor(1000*1.04719755...) = floor(1047.19755...) so a(1000) = 1047.

Paulo Romero Zanconato Pinto, Table of n, a(n) for n = 0..10000
Index entries for sequences related to Beatty sequences

Complement of A277051.

Cf. A000796, A019670.

A274399:=n->floor(n*Pi/3): seq(A274399(n), n=0..100); # Wesley Ivan Hurt, Sep 26 2016

f[n_] := Floor[n*Pi/Floor[Pi]]; Array[f, 100, 0]

Maxima
```
makelist(floor(n*(%pi/3)), n, 0, 100);
```

a(n) = n*Pi\3; \\ Michel Marcus, Sep 26 2016

a(n) = floor(n*Pi/floor(Pi)) = floor(n*Pi/3).

A277051 a(n) = floor(n/(1-3/Pi)).

Original entry on oeis.org

22, 44, 66, 88, 110, 133, 155, 177, 199, 221, 244, 266, 288, 310, 332, 355, 377, 399, 421, 443, 465, 488, 510, 532, 554, 576, 599, 621, 643, 665, 687, 710, 732, 754, 776, 798, 820, 843, 865, 887, 909, 931, 954, 976, 998, 1020, 1042, 1065, 1087, 1109, 1131, 1153, 1175

Offset: 1

Paulo Romero Zanconato Pinto, Sep 26 2016

			For n = 10 we have that floor(10/(1-3/Pi)) = floor(10/0.04719755...) = floor(211.8754045...) so a(10) = 221.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Index entries for sequences related to Beatty sequences

Complement of A274399.

Cf. A000796, A019670.

A277051:=n->floor(n/(1-3/Pi)): seq(A277051(n), n=1..100); # Wesley Ivan Hurt, Sep 26 2016

f[n_] := Floor[n/(1-3/Pi)]; Array[f, 100, 1]

makelist(floor(n / (1-3 / %pi )), n, 1, 100);

a(n)=n\(1-3/Pi) \\ Charles R Greathouse IV, Sep 26 2016

a(n) = floor(n/(1-3/Pi)).

A325742 First term of n-th difference sequence of (floor(Pi*k/3)), k >= 0.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -22, 253, -2024, 12650, -65780, 296010, -1184040, 4292145, -14307150, 44352165, -129024480, 354817320, -927983760, 2319959400, -5567902560, 12875774670, -28781143380, 62359143990

Offset: 1

Clark Kimberling, Jun 05 2019

Clark Kimberling, Table of n, a(n) for n = 1..200

Cf. A019670, A325664.

Table[First[Differences[Table[Floor[(Pi/3)*n], {n, 0, 50}], n]], {n, 1, 50}]

A336077 Decimal expansion of (10Pi + 3sqrt(3)) / (2Pi - 3sqrt(3)).

Original entry on oeis.org

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

Offset: 2

Clark Kimberling, Jul 11 2020

Decimal expansion of the ratio of segment areas for arclength Pi/3 on the unit circle. In general, suppose that s in (0,Pi) is the length of an arc of the unit circle. The associated chord separates the interior into two segments. Let A1 be the area of the larger and A2 the area of the smaller. The term "ratio of segment areas" means A1/A2. See A336073 for a guide to related sequences.

			33.68074644435052842991251795285920080736045858...

Index entries for transcendental numbers

Cf. A336073.

s := Pi/3 ;
sss := s-sin(s) ;
evalf( 2*Pi/sss -1 ) ; # R. J. Mathar, Sep 02 2020

s = Pi/3; r = N[(2 Pi - s + Sin[s])/(s - Sin[s]), 200]
RealDigits[r][[1]]

Equals (2*Pi - s + sin(s))/(s - sin(s)), where s = Pi/3 = A019670.

cp's OEIS Frontend

A204067 Decimal expansion of the Fresnel Integral, Integral_{x >= 0} cos(x^3) dx.

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A238238 Decimal expansion of the polar angle, in radians, of a cone which makes a golden-ratio cut of the full solid angle.

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A352453 Decimal expansion of the area of intersection of 4 unit-radius circles that have the vertices of a unit-side square as centers.

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A281452 Expansion of f(x, x) * f(x^5, x^13) in powers of x where f(, ) is Ramanujan's general theta function.

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A346573 Decimal expansion of 2 - Pi/3.

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A152627 Decimal expansion of 3/4.

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A274399 a(n) = floor(n*Pi/3).

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A336077 Decimal expansion of (10Pi + 3sqrt(3)) / (2Pi - 3sqrt(3)).