A244978 -id:A244978 - cp's OEIS frontend

A019675 Decimal expansion of Pi/8.

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

Offset: 0

N. J. A. Sloane

Equals Integral_{x>=0} sin(4*x)/(4*x) dx. - Jean-François Alcover, Feb 28 2013

Consider 4 circles inscribed in a square. Inscribe a square in each circle. And finally, inscribe 4 circles inside each four small squares. Totally we get 16 small circles. Pi/8 is the ratio of the area of the 16 small circles to the area of initial square. See the link. - Kirill Ustyantsev, Apr 30 2020

			Pi/8 = 0.392699081698724154807830422909937860524646174921888227621868... - _Vladimir Joseph Stephan Orlovsky_, Dec 02 2009

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.4, p. 492.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Kirill Ustyantsev, Geometric sense of Pi/8.
Index entries for transcendental numbers.

Cf. A195909, A195913, A195697. - Mohammad K. Azarian, Oct 11 2011
Cf. A000796, A003881, A019670, A024235, A244978, A334422 (Pi/128).
Cf. A001539, A061550.

pi:=Pi(RealField(110)); Reverse(Intseq(Floor(10^100*(pi)/8))); // Vincenzo Librandi, Oct 07 2015

RealDigits[N[Pi/8,6! ]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 *)

default(realprecision, 1002);
eval(vecextract(Vec(Str(sumalt(n=0, (-1)^(n)/(4*n+2)))), "3..-2"))  \\ Gheorghe Coserea, Oct 06 2015

From Peter Bala, Nov 15 2016: (Start)
Pi/8 = Sum_{k >= 1} (-1)^k/((2*k - 3)*(2*k - 1)*(2*k + 1)).
More generally, for n >= 0 we have 1/(2*n)! * Pi/4 = Sum_{k >= 1} (-1)^(k+n-1) * 1/Product_{j = -n..n} (2*k + 2*j - 1): when n = 0 we get the Madhava-Gregory-Leibniz series for Pi/4.
For N divisible by 4, we have the asymptotic expansion Pi/8 - Sum_{k = 1..N/2} (-1)^k/((2*k - 3)*(2*k - 1)*(2*k + 1)) ~ -1/2*(1/N^3 - 2/N^5 + 31/N^7 - 692/N^9 + ...), where the sequence of unsigned coefficients [1, 2, 31, 692, ...] equals A024235. (End)
Equals Integral_{x = 0..1} x*sqrt(1 - x^4) dx. - Peter Bala, Oct 27 2019
Equals Integral_{x = 0..oo} sin(x)^6/x^4 dx = Sum_{n >= 1} sin(n)^6/n^4, by the Abel-Plana formula. - Peter Bala, Nov 04 2019
From Amiram Eldar, Jul 12 2020: (Start)
Equals arctan(sqrt(2) - 1).
Equals Sum_{k>=0} (-1)^k/(4*k+2).
Equals Sum_{k>=0} 1/((4*k+1)*(4*k+3)) = Sum_{k>=0} 1/A001539(k).
Equals Integral_{x=0..oo} dx/(x^2 + 16).
Equals Integral_{x=0..oo} dx/(x^4 + 4) = Integral_{x=0..oo} x/(x^4 + 4) dx.
Equals Integral_{x=0..oo} x/(x^4 + 1)^2 dx = Integral_{x=0..1} x/(x^4 + 1) dx.
Equals Integral_{x=0..1} x * arcsin(x) dx. (End)
From Kritsada Moomuang, Jun 18 2025: (Start)
Equals Integral_{x=0..oo} (x*log(x + 1))/((x^2 + 1)^2) dx.
Equals Integral_{x=0..oo} (x^3 - 3*x + 3*arctan(x))/(3*x^5) dx. (End)

A051062 a(n) = 16*n + 8.

Original entry on oeis.org

8, 24, 40, 56, 72, 88, 104, 120, 136, 152, 168, 184, 200, 216, 232, 248, 264, 280, 296, 312, 328, 344, 360, 376, 392, 408, 424, 440, 456, 472, 488, 504, 520, 536, 552, 568, 584, 600, 616, 632, 648, 664, 680, 696, 712, 728, 744, 760, 776, 792, 808, 824, 840

Offset: 0

N. J. A. Sloane, Gary W. Adamson, Dec 11 1999

Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0(97).
n such that 32 is the largest power of 2 dividing A003629(k)^n-1 for any k. - Benoit Cloitre, Mar 23 2002
Continued fraction expansion of tanh(1/8). - Benoit Cloitre, Dec 17 2002
If Y and Z are 2-blocks of a (4n+1)-set X then a(n-1) is the number of 3-subsets of X intersecting both Y and Z. - Milan Janjic, Oct 28 2007
General form: (q*n+x)*q x=+1; q=2=A016825, q=3=A017197, q=4=A119413, ... x=-1; q=3=A017233, q=4=A098502, ... x=+2; q=4=A051062, ... - Vladimir Joseph Stephan Orlovsky, Feb 16 2009
a(n)*n+1 = (4n+1)^2 and a(n)*(n+1)+1 = (4n+3)^2 are both perfect squares. - Carmine Suriano, Jun 01 2014
For all positive integers n, there are infinitely many positive integers k such that k*n + 1 and k*(n+1) + 1 are both perfect squares. Except for 8, all the numbers of this sequence are the smallest integers k which are solutions for getting two perfect squares. Example: a(1) = 24 and 24 * 1 + 1 = 25 = 5^2, then 24 * (1+1) + 1 = 49 = 7^2. [Reference AMM] - Bernard Schott, Sep 24 2017
Numbers k such that 3^k + 1 is divisible by 17*193. - Bruno Berselli, Aug 22 2018

Letter from Gary W. Adamson concerning Prouhet-Thue-Morse sequence, Nov 11 1999.

Mia Boudreau, Table of n, a(n) for n = 0..10000
Mihaly Bencze, Problem 11508, The American Mathematical Monthly, Vol. 117, N° 5, May 2010, p. 459.
Milan Janjić, Two Enumerative Functions. [Wayback Machine link]
Tanya Khovanova, Recursive Sequences.
William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_0(N)).
William A. Stein, The modular forms database.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A005408, A008598, A017113, A106839, A139098, A244978.
Cf. A003484, A007814, A037227, A118413, A209675.
Cf. A003629, A016825, A017197, A017233, A098502, A119413.

[16*n+8: n in [0..50]]; // Wesley Ivan Hurt, Jun 01 2014

A051062:=n->16*n+8; seq(A051062(n), n=0..50); # Wesley Ivan Hurt, Jun 01 2014

Range[8, 1000, 16] (* Vladimir Joseph Stephan Orlovsky, May 31 2011 *)
Table[16n+8, {n,0,50}] (* Wesley Ivan Hurt, Jun 01 2014 *)
LinearRecurrence[{2,-1},{8,24},60] (* or *) NestList[#+16&,8,60] (* Harvey P. Dale, Aug 18 2019 *)

a(n)=16*n+8 \\ Charles R Greathouse IV, May 09 2016

a(n) = A118413(n+1,4) for n>3. - Reinhard Zumkeller, Apr 27 2006
a(n) = 32*n - a(n-1) for n>0, a(0)=8. - Vincenzo Librandi, Aug 06 2010
A003484(a(n)) = 8; A209675(a(n)) = 9. - Reinhard Zumkeller, Mar 11 2012
A007814(a(n)) = 3; A037227(a(n)) = 7. - Reinhard Zumkeller, Jun 30 2012
a(-1 - n) = - a(n). - Michael Somos, Jun 02 2014
Sum_{n>=0} (-1)^n/a(n) = Pi/32 (A244978). - Amiram Eldar, Feb 28 2023
From Elmo R. Oliveira, Apr 16 2024: (Start)
G.f.: 8*(1+x)/(1-x)^2.
E.g.f.: 8*exp(x)*(1 + 2*x).
a(n) = 8*A005408(n) = A008598(n) + 8 = A139098(n+1) - A139098(n).
a(n) = 4*A016825(n) = 2*A017113(n) = 2*a(n-1) - a(n-2) for n >= 2. (End)
From Amiram Eldar, Nov 25 2024: (Start)
Product_{n>=0} (1 - (-1)^n/a(n)) = sqrt(2)*sin(7*Pi/32).
Product_{n>=0} (1 + (-1)^n/a(n)) = sqrt(2)*cos(7*Pi/32). (End)

A019679 Decimal expansion of Pi/12.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Equals cone's volume (radius = 1/2, height = 1) and semi-sphere's volume (radius = 1/2). - Eric Desbiaux, Dec 08 2008
Decimal expansion of least x > 0 having cos(4x) = (cos 3x)^2. See A197476. - Clark Kimberling, Oct 15 2011
Multiplied by 10, decimal expansion of 5*Pi/6. - Alonso del Arte, Aug 19 2013
Volume between a cylinder and the inscribed sphere of diameter 1. - Omar E. Pol, Sep 25 2013

			Pi/12 = 0.2617993877991494365385536152732919070164307...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.4, p. 492.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Index entries for transcendental numbers.

Cf. A000796, A003881, A019669, A019670, A019675, A019683, A244978.

RealDigits[N[Pi/12, 6! ]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 *)
RealDigits[Pi/12,10,120][[1]] (* Harvey P. Dale, Jan 12 2024 *)

Pi/12 \\ Charles R Greathouse IV, Sep 28 2022

A003881 - A019673. - Omar E. Pol, Sep 25 2013
Equals Integral_{x = 0..1} x^2*sqrt(1 - x^6) dx. - Peter Bala, Oct 27 2019
Equals Sum_{k>=0} binomial(2*k,k)/((2*k+1)*4^(2*k+1)). - Amiram Eldar, May 30 2021
Constant divided by 10 = Pi/120 = 0.0261799387... =  Sum_{n = -oo..oo} 1/((4*n+1)*(4*n+2)*(4*n+3)*(4*n+5)*(4*n+6)*(4*n+7)) (using the Eisenstein summation convention Sum_{n = -oo..oo} = lim_{N -> oo} Sum_{n = -N..N}). Note that 22/7 - Pi = 240*Sum_{n >= 1} 1/((4*n+1)*(4*n+2)*(4*n+3)*(4*n+5)*(4*n+6)*(4*n+7)). - Peter Bala, Nov 28 2021

A019683 Decimal expansion of Pi/16.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

			Pi/16 = 0.19634954084936207740391521145496893026232308746094411381... - _Vladimir Joseph Stephan Orlovsky_, Dec 02 2009

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.4.2, p. 494.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Index entries for transcendental numbers.

Cf. A157332, A000796, A003881, A019669, A019670, A019675, A019679, A244978.

R:= RealField(100); Pi(R)/16; // G. C. Greubel, Aug 26 2019

RealDigits[N[Pi/16,6! ]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 *)

default(realprecision, 100); Pi/16 \\ G. C. Greubel, Aug 26 2019

numerical_approx(pi/16, digits=100) # G. C. Greubel, Aug 26 2019

From  Peter Bala, Oct 27 2019: (Start)
Equals Integral_{x = 0..1} x^2*sqrt(1 - x^2) dx = Integral_{x = 0..1} x^3*sqrt(1 - x^8) dx.
Equals Integral_{x = 0..inf} x^2/(1 + x^2)^3 dx. (End)
From Amiram Eldar, Aug 04 2020: (Start)
Equals Sum_{k>=1} sin(k)^3 * cos(k)/k.
Equals Sum_{k>=1} sin(k)^3 * cos(k)^2/k.
Equals Sum_{k>=1} (-1)^(k+1) * sin((2*k-1)/4)/(2*k-1)^2. (End)

A244979 Decimal expansion of Pi/(2*sqrt(5)).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jul 09 2014

			0.702481473104072639315637464320489479946650918706720241998972102619214188...

George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), Chapter 13 A Master Formula, p. 250.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Beta Function
Index entries for transcendental numbers

Cf. A244976, A244977, A244978, A019692.

RealDigits[Pi/(2*Sqrt[5]), 10, 105] // First

Pi/sqrt(20) \\ Charles R Greathouse IV, Sep 30 2022

Equals Integral_(0..1) (1 + x^2)/(1 + 3*x^2 + x^4) dx.
From Peter Bala, Feb 16 2015: (Start)
Also equals beta(1/2, 1/2)/(2*sqrt(5)), where 'beta' is Euler's beta function.
Pi/(2*sqrt(5)) = Integral_{t = 0..a} (1 + t^2)*(1 + t^6)/(1 + t^10) dt = a + a^3/3 + a^7/7 + a^9/9 - a^11/11 - a^13/13 - a^17/17 - a^19/19 + ..., where a = 1/2(sqrt(5) - 1). Hint: differentiate atan( sqrt(5)*(t - t^3)/(1 - 3*t^2 + t^4) ). (End)
Equals (1/2)*Sum_{n >= 0} (-1)^n*( 1/(10*n + 1) + 1/(10*n + 3) + 1/(10*n + 7) + 1/(10*n + 9) ). Cf. A019692. - Peter Bala, Oct 30 2019
From Amiram Eldar, Aug 06 2020: (Start)
Equals Integral_{x=0..oo} 1/(x^2 + 5) dx.
Equals 0.1 * Integral_{x=0..oo} log(1 + 5/x^2) dx. (End)
Equals Integral_{x = 0..1} 2/(4*x^2 + 5*(1 - x)^2) dx. - Peter Bala, Jul 22 2022

A244980 Decimal expansion of Pi/(2*sqrt(6)).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jul 09 2014

			0.6412749150809320477720181798355032057336303337820461615506948033781994...

George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), Chapter 13 A Master Formula, p. 250.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Beta Function
Index entries for transcendental numbers

Cf. A244976, A244977, A244978, A244979.

RealDigits[Pi/(2*Sqrt[6]), 10, 106] // First

Pi/sqrt(24) \\ Charles R Greathouse IV, Oct 01 2022

Equals Integral_{x=0..1} (1 + x^2)/(1 + 4*x^2 + x^4) dx.
Equals beta(1/2, 1/2)/(2*sqrt(6)), where 'beta' is Euler's beta function.
From Amiram Eldar, Aug 15 2020: (Start)
Equals Integral_{x=0..oo} 1/(x^2 + 6) dx.
Equals Integral_{x=0..oo} 1/(2*x^2 + 3) dx.
Equals Integral_{x=0..oo} 1/(3*x^2 + 2) dx.
Equals Integral_{x=0..oo} 1/(6*x^2 + 1) dx. (End)
Equals Integral_{x = 0..1} 1/(2*x^2 + 3*(1 - x)^2) dx. - Peter Bala, Jul 22 2022

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A019675 Decimal expansion of Pi/8.

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A051062 a(n) = 16*n + 8.

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A019679 Decimal expansion of Pi/12.

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A019683 Decimal expansion of Pi/16.

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A244979 Decimal expansion of Pi/(2*sqrt(5)).

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A244980 Decimal expansion of Pi/(2*sqrt(6)).

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