A153799 -id:A153799 - cp's OEIS frontend

A074378 Even triangular numbers halved.

0, 3, 5, 14, 18, 33, 39, 60, 68, 95, 105, 138, 150, 189, 203, 248, 264, 315, 333, 390, 410, 473, 495, 564, 588, 663, 689, 770, 798, 885, 915, 1008, 1040, 1139, 1173, 1278, 1314, 1425, 1463, 1580, 1620, 1743, 1785, 1914, 1958, 2093, 2139, 2280, 2328, 2475

Offset: 0

W. Neville Holmes, Sep 04 2002

Set of integers k such that k + (1 + 2 + 3 + 4 + ... + x) = 3*k, where x is sufficiently large. For example, 203 is a term because 203 + (1 + 2 + 3 + 4 + ... +28) = 609 and 609 = 3*203. - Gil Broussard, Sep 01 2008
Set of all m such that 16*m+1 is a perfect square. - Gary Detlefs, Feb 21 2010
Integers of the form Sum_{k=0..n} k/2. - Arkadiusz Wesolowski, Feb 07 2012
Numbers of the form h*(4*h + 1) for h = 0, -1, 1, -2, 2, -3, 3, ... - Bruno Berselli, Feb 26 2018
Numbers whose distance to nearest square equals their distance to nearest oblong; that is, numbers k such that A053188(k) = A053615(k). - Lamine Ngom, Oct 27 2020
The sequence terms are the exponents in the expansion of Product_{n >= 1} (1 - q^(8*n))*(1 + q^(8*n-3))*(1 + q^(8*n-5)) = 1 + q^3 + q^5 + q^14 + q^18 + .... - Peter Bala, Dec 30 2024

David A. Corneth, Table of n, a(n) for n = 0..9999
Neville Holmes, More Geometric Integer Sequences
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A011848, A014493, A014494, A074377, A035294, A274757.
Cf. A010709, A047522. [Vincenzo Librandi, Feb 14 2009]
Cf. A266883 (numbers n such that 16*n-15 is a square).
Cf. A016687, A153799.
Cf. A053615, A053188.

f:=func; [0] cat [f(n*m): m in [-1,1], n in [1..25]]; // Bruno Berselli, Nov 13 2012

Maple

a:=n->(2*n+1)*floor((n+1)/2): seq(a(n),n=0..50); # Muniru A Asiru, Feb 01 2019

Mathematica

1/2 * Select[PolygonalNumber@ Range[0, 100], EvenQ] (* Michael De Vlieger, Jun 01 2017, Version 10.4 *) Select[Accumulate[Range[0,100]],EvenQ]/2 (* Harvey P. Dale, Feb 15 2025 *)

PARI

a(n)=(2*n+1)*(n-n\2)

Formula

Sum_{n>=0} q^a(n) = (Prod_{n>0} (1-q^n))*(Sum_{n>=0} A035294(n)*q^n).

a(n) = n*(n + 1)/4 where n*(n + 1)/2 is even.

G.f.: x*(3 + 2*x + 3*x^2)/((1 - x)*(1 - x^2)^2).

From Benoit Jubin, Feb 05 2009: (Start)

a(n) = (2*n + 1)*floor((n + 1)/2).

a(2*k) = k*(4*k+1); a(2*k+1) = (k+1)*(4*k+3). (End)

a(2*n) = A007742(n), a(2*n-1) = A033991(n). - Arkadiusz Wesolowski, Jul 20 2012

a(n) = (4*n + 1 - (-1)^n)*(4*n + 3 - (-1)^n)/4^2. - Peter Bala, Jan 21 2019

a(n) = (2*n+1)*(n+1)*(1+(-1)^(n+1))/4 + (2*n+1)*(n)*(1+(-1)^n)/4. - Eric Simon Jacob, Jan 16 2020

From Amiram Eldar, Jul 03 2020: (Start)

Sum_{n>=1} 1/a(n) = 4 - Pi (A153799).

Sum_{n>=1} (-1)^(n+1)/a(n) = 6*log(2) - 4 (See A016687). (End)

a(n) = A014494(n)/2 = A274757(n)/3 = A266883(n) - 1. - Hugo Pfoertner, Dec 31 2024

A063448 Decimal expansion of Pi * sqrt(2).

Original entry on oeis.org

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

Offset: 1

Jason Earls, Jul 24 2001

Hypotenuse of the right triangle with legs Pi and Pi. - Zak Seidov, May 04 2005
Circumference of the circumcircle of the unit square. - Jonathan Sondow, Nov 23 2017
Half-perimeter of the closed curve with implicit Cartesian equation x^2 + y^2 = abs(x) + abs(y). - Stefano Spezia, Oct 20 2020

			4.4428829381583662470158809900606936986146216893756902230853...

Harry J. Smith, Table of n, a(n) for n = 1..20000
Peter Bala, Series for sqrt(2)*Pi
Eric Weisstein's World of Mathematics, Gauss Multiplication Formula.
Wikipedia, Bailey-Borwein-Plouffe formula.
Index entries for transcendental numbers

Cf. A063447 (continued fraction), A093954, A153799, A193887, A244976, A247719.

RealDigits[N[Pi*Sqrt[2], 200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Mar 21 2011*)

PARI
```
\p 400; Pi * sqrt(2)
```

default(realprecision, 20080); x=Pi*sqrt(2); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b063448.txt", n, " ", d)) \\ Harry J. Smith, Aug 21 2009

# Use some guard digits when computing.
# BBP formula (1/8) P(1, 64, 12, (32, 0, 8, 0, 8, 0, -4, 0, -1, 0, -1, 0))
from decimal import Decimal as dec, getcontext
def BBPpisqrt2(n: int) -> dec:
    getcontext().prec = n
    s = dec(0); f = dec(1); g = dec(64)
    for k in range(int(n * 0.5536546824812272) + 1):
        twk = dec(12 * k)
        s += f * ( dec(32) / (twk + 1) + dec(8)  / (twk + 3)
                 + dec(8)  / (twk + 5) - dec(4)  / (twk + 7)
                 - dec(1)  / (twk + 9) - dec(1)  / (twk + 11))
        f /= g
    return s / dec(8)
print(BBPpisqrt2(200))  # Peter Luschny, Nov 03 2023

Equals Gamma(1/4)*Gamma(3/4). - Jean-François Alcover, Nov 24 2014
From Amiram Eldar, Aug 06 2020: (Start)
Equals Integral_{x=0..oo} log(1 + 1/x^4) dx.
Equals Integral_{x=0..oo} log(1 + 2/x^2) dx.
Equals Integral_{x=-oo..oo} exp(x/4)/(exp(x) + 1) dx.
Equals Integral_{x=0..2*Pi} 1/(cos(x)^2 + 1) dx = Integral_{x=0..2*Pi} 1/(sin(x)^2 + 1) dx. (End)
From Andrea Pinos, Jul 03 2023: (Start)
Equals (Product_{k=1..4} Gamma(k/8)*Gamma(1 - k/8))^(1/4).
General result: 2*Pi/(4*y)^(1/(2*y)) = (Product_{k=1..y} Gamma(k/(2*y))*Gamma(1 - k/(2*y)) )^(1/y). (End)
From Peter Bala, Oct 22 2023: (Start)
sqrt(2)*Pi = 4 + 8*Sum_{n >= 0} (-1)^n/(16*n^2 + 32*n + 15). See A141759.
In the following the Eisenstein summation convention is assumed: that is,
Sum_{n = -oo..oo} means Limit_{N -> oo} Sum_{n = -N..N}:
sqrt(2)*Pi = 4*Sum_{n = -oo..oo} (-1)^n/(4*n + 1).
More generally, it appears that for k >= 0, k not of the form 4*m + 1,
sqrt(2)*Pi = -sign(cos(Pi*(k - 3)/4)) * 4*(2^floor(k/2))*k! * Sum_{n = -oo..oo} (-1)^n/((4*n + 1)*(4*n + 3)*...*(4*n + 2*k + 1)) (verified up to k = 50).
sqrt(2)*Pi = (2^4)*Sum_{n >= 0} (-1)^n * (2*n + 1)/((4*n + 1)*(4*n + 3)) = 512/105 - (2^6)*4!*Sum_{n >= 0} (-1)^n * (2*n + 3)/((4*n + 1)*(4*n + 3)*...*(4*n + 11)).
sqrt(2)*Pi = 4 + (2^3)*Sum_{n >= 0} (-1)^n * (4*n + 1)/((4*n + 1)*(4*n + 3)*(4*n + 5)) = 1408/315 - (2^5)*5!*Sum_{n >= 0} (-1)^n * (4*n + 1)/((4*n + 1)*(4*n + 3)*...*(4*n + 13)).
sqrt(2)*Pi = 16/3 - (2^4)*3!*Sum_{n >= 0} (-1)^n/((4*n + 1)*(4*n + 3)*(4*n + 5)*(4*n + 7)) = 14848/3465 + (2^6)*7!*Sum_{n >= 0} (-1)^n/((4*n + 1)*(4*n + 3)*...*(4*n + 15)). (End)
From Peter Bala, Nov 19 2023: (Start)
sqrt(2)*Pi = 512*Sum_{k >= 1} (-1)^(k+1) * k^2/((16*k^2 - 1)*(16*k^2 - 9)).
This is the case n = 1 of the more general result: for n >= 1,
sqrt(2)*Pi = (-1)^(n+1) * 2^(n+7) * (2*n)!/(2*n - 1) * Sum_{k >= 1} (-1)^(k+1) * k^2/( Product_{i = 0..n} (16*k^2 - (2*i+1)^2) ). Cf. A334422. (End)
Equals Integral_{x=-oo..oo} (x^2 + 1)/(x^4 + 1) dx. - Kritsada Moomuang, Jun 04 2025

Edited by N. J. A. Sloane, May 05 2007

Corrected by Neven Juric, Apr 24 2008

A180434 Decimal expansion of constant (2 - Pi/2).

Original entry on oeis.org

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

Offset: 0

Jonathan Vos Post, Sep 05 2010

(2-Pi/2)*a^2 is the area of the loop of the right strophoid (also called the Newton strophoid) whose polar equation is r = a*cos(2*t)/cos(t) and whose Cartesian equation is x*(x^2+y^2) = a*(x^2-y^2) or y = +- x*sqrt((a-x)/(a+x)). See the curve with its loop at the Mathcurve link; the loop appears for -Pi/4 <= t <= Pi/4. - Bernard Schott, Jan 28 2020

			0.42920367320510338076867830836024855790141530...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Robert Ferréol, Right strophoid, Math Curve.
Nikita Kalinin and Mikhail Shkolnikov, The number Pi and summation by SL(2,Z), arXiv:1701.07584 [math.NT], 2016. Gives a formula.
Index entries for transcendental numbers

Cf. A004601, A153799, A180433, A222362.

RealDigits[2-Pi/2,10,120][[1]] (* Harvey P. Dale, Oct 12 2013 *)

Equals Integral_{t=0..Pi/4} ((cos(2*t))/cos(t))^2 dt. - Bernard Schott, Jan 28 2020
From Amiram Eldar, May 30 2021: (Start)
Equals Sum_{k>=1} 2^k/(binomial(2*k,k)*k*(2*k + 1)).
Equals Integral_{x=0..1} arcsin(x)*arccos(x) dx. (End)
Equals Integral_{x=0..1} sqrt(x)/(1+x) dx. - Andy Nicol, Mar 23 2024
Equals A153799/2. - Hugo Pfoertner, Mar 23 2024

Corrected by Carl R. White, Sep 09 2010

More terms from N. J. A. Sloane, Sep 23 2010

A210958 Decimal expansion of 1 - (Pi/4).

Original entry on oeis.org

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

Offset: 0

Omar E. Pol, Aug 02 2012

Decimal expansion of (4 - Pi)/4.
Area between a square and the inscribed quarter circle of radius 1.
Also area between a circle of radius 1 and the circumscribed square, divided by 4.
Also area between a circle of diameter 1 and the circumscribed square. - Omar E. Pol, Sep 24 2013
Also volume between a cube of side length 1 and the inscribed cylinder. - Omar E. Pol, Sep 25 2013

			0.21460183660255169038433915418012427895070765015622...

M. L. Glasser, A note on Beukers's and related integrals, Amer. Math. Monthly 126(4) (2019), 361-363.
Index entries for transcendental numbers

Cf. A003881, A153799.

Essentially the same as A091651.

RealDigits[1 - Pi/4, 10, 87][[1]] (* Bruno Berselli, Aug 03 2012 *)

1-Pi/4 \\ Charles R Greathouse IV, Oct 01 2022

1 - (Pi/4) = (4 - Pi)/4 = 1 - A003881 = A153799/4.
From Amiram Eldar, Jun 29 2020: (Start)
Equals Sum_{k>=0} (-1)^k/(2*k+3).
Equals Integral_{x=0..Pi/4} tan(x)^2 dx.
Equals Integral_{x=0..1} arcsin(x) dx /(1+x)^2.
Equals Integral_{x=1..oo} dx/(x^2+x^4). (End)
Equals -Integral_{x=0..1, y=0..1} arcsin(x*y)/((1+x*y)^2*log(x*y)) dx dy. (Apply Theorem 1 or Theorem 2 from Glasser (2019) to one of Amiram Eldar's integrals.) - Petros Hadjicostas, Jun 29 2020
Continued fraction 1/(3 + 3^2/(2 + 5^2/(2 + 7^2/(2 + ... )))). - Peter Bala, Feb 28 2024

More terms from David Scambler, Aug 02 2012

A210962 Decimal expansion of 4*(2 - Pi/3).

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 03 2012

Volume between a sphere of radius 1 and the circumscribed cube.

			3.8112097952136090153831...

Index entries for transcendental numbers

Cf. A019670, A019699, A153799.

RealDigits[4 (2 - Pi/3), 10, 87][[1]] (* Bruno Berselli, Aug 03 2012 *)

8 - 4/3*Pi \\ Charles R Greathouse IV, Oct 01 2022

4*(2 - Pi/3) = 8 - 4*Pi/3 = 8 - A019699.

cp's OEIS Frontend

A074378 Even triangular numbers halved.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A063448 Decimal expansion of Pi * sqrt(2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Python

Formula

Extensions

A180434 Decimal expansion of constant (2 - Pi/2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A210958 Decimal expansion of 1 - (Pi/4).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A210962 Decimal expansion of 4*(2 - Pi/3).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula