A019683 -id:A019683 - cp's OEIS frontend

A017113 a(n) = 8*n + 4.

4, 12, 20, 28, 36, 44, 52, 60, 68, 76, 84, 92, 100, 108, 116, 124, 132, 140, 148, 156, 164, 172, 180, 188, 196, 204, 212, 220, 228, 236, 244, 252, 260, 268, 276, 284, 292, 300, 308, 316, 324, 332, 340, 348, 356, 364, 372, 380, 388, 396, 404, 412, 420, 428, 436, 444, 452, 460, 468

Offset: 0

N. J. A. Sloane

Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0(65).
n such that 16 is the largest power of 2 dividing A003629(k)^n - 1 for any k. - Benoit Cloitre, Mar 23 2002
Continued fraction expansion of tanh(1/4). - Benoit Cloitre, Dec 17 2002
Consider all primitive Pythagorean triples (a,b,c) with c - a = 8, sequence gives values for b. (Corresponding values for a are A078371(n), while c follows A078370(n).) - Lambert Klasen (Lambert.Klasen(AT)gmx.net), Nov 19 2004
Also numbers of the form a^2 + b^2 + c^2 + d^2, where a,b,c,d are odd integers. - Alexander Adamchuk, Dec 01 2006
If X is an n-set and Y_i (i=1,2,3) mutually disjoint 2-subsets of X then a(n-5) is equal to the number of 4-subsets of X intersecting each Y_i (i=1,2,3). - Milan Janjic, Aug 26 2007
A007814(a(n)) = 2; A037227(a(n)) = 5. - Reinhard Zumkeller, Jun 30 2012
Numbers k such that 3^k + 1 is divisible by 41. - Bruno Berselli, Aug 22 2018
Lexicographically smallest arithmetic progression of positive integers avoiding Fibonacci numbers. - Paolo Xausa, May 08 2023
From Martin Renner, May 24 2024: (Start)
Also number of points in a grid cross with equally long arms and a width of two points, e.g.:
                                         * *
                        * *              * *
           * *          * *              * *
  * *    * * * *    * * * * * *    * * * * * * * *
  * *    * * * *    * * * * * *    * * * * * * * *
           * *          * *              * *
                        * *              * *
                                         * *
etc. (End)

Paolo Xausa, Table of n, a(n) for n = 0..10000 (terms 0..1100 from Vincenzo Librandi)
E. Catalan, Extrait d'une lettre, Bulletin de la S. M. F., tome 17 (1889), pp. 205-206. [If N is a prime number of the form 4*m+1, then 8*N+4 is the sum of four odd squares.]
Cody Clifton, Commutativity in non-Abelian Groups, May 06 2010.
Colin Defant and Noah Kravitz, Loops and Regions in Hitomezashi Patterns, arXiv:2201.03461 [math.CO], 2022. Theorem 1.2.
Dr Barker, How to Avoid the Fibonacci Numbers, YouTube video, 2023.
Meimei Gu and Rongxia Hao, 3-extra connectivity of 3-ary n-cube networks, arXiv:1309.5083 [cs.DM], Sep 19, 2013.
Milan Janjic, Two Enumerative Functions.
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).

First differences of A016742 (even squares).
Cf. A078370, A078371, A081770 (subsequence).
Cf. A003629, A005408, A007814, A019683, A037227, A051062, A118413.
Cf. A008586, A016825.

a017113 = (+ 4) . (* 8)
a017113_list = [4, 12 ..]  -- Reinhard Zumkeller, Jul 13 2013

[8*n+4: n in [0..50]]; // Vincenzo Librandi, Apr 26 2011

LinearRecurrence[{2,-1}, {4,12}, 50] (* G. C. Greubel, Apr 26 2018 *)

a(n)=8*n+4 \\ Charles R Greathouse IV, Sep 23 2013

a(n) = A118413(n+1,3) for n > 2. - Reinhard Zumkeller, Apr 27 2006
a(n) = Sum_{k=0..4*n} (i^k+1)*(i^(4*n-k)+1), where i = sqrt(-1). - Bruno Berselli, Mar 19 2012
a(n) = 4*A005408(n). - Omar E. Pol, Apr 17 2016
E.g.f.: (8*x + 4)*exp(x). - G. C. Greubel, Apr 26 2018
G.f.: 4*(1+x)/(1-x)^2. - Wolfdieter Lang, Oct 27 2020
Sum_{n>=0} (-1)^n/a(n) = Pi/16 (A019683). - Amiram Eldar, Dec 11 2021
From Amiram Eldar, Nov 22 2024: (Start)
Product_{n>=0} (1 - (-1)^n/a(n)) = sqrt(2) * sin(3*Pi/16).
Product_{n>=0} (1 + (-1)^n/a(n)) = sqrt(2) * cos(3*Pi/16). (End)
a(n) = 2*A016825(n) = A008586(2*n+1). - Elmo R. Oliveira, Apr 10 2025

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

A244978 Decimal expansion of Pi/32.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jul 09 2014

			0.0981747704246810387019576057274844651311615437304720569054670185096...

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 MathWorld, Beta Function
Eric W. Weisstein, Euler's Series Transformation.
Index entries for transcendental numbers

Cf. A019683, A244976, A244977, A019675, A000796, A003881, A019669, A019670, A019683.

Join[{0}, RealDigits[Pi/32, 10, 105] // First]

Pi/32 \\ Charles R Greathouse IV, Sep 28 2022

Equals Integral_{x = 0..1} x^2/(1 + x^2)^3 dx.
Also equals beta(3/2, 1/2)/16, where 'beta' is Euler's beta function.
From Peter Bala, Oct 27 2019: (Start)
Equals Integral_{x = 0..1} x^4*sqrt(1 - x^2) dx = Integral_{x = 0..1} x^5*sqrt(1 - x^4) dx = Integral_{x = 0..1} x^7*sqrt(1 - x^16) dx.
Equals Integral_{x >= 0} x^4/(1 + x^2)^4 dx. (End)
From Amiram Eldar, Jul 13 2020: (Start)
Equals Integral_{x=0..oo} dx/(x^2 + 4)^2.
Equals Sum_{k>=1} sin(k)^3*cos(k)^3/k. (End)
From  Peter Bala, Dec 08 2021: (Start)
Pi/32 = Sum_{n >= 1} (-1)^n*n^2/((4*n^2 - 1)*(4*n^2 - 9)).
Applying Euler's series transformation to this alternating sum gives
Pi/32 = Sum_{n >= 1} 2^(n-3)*n*(n+1)/((2*n+3)*binomial(2*n+2, n+1)). (End)

A136483 Number of unit square lattice cells inside quadrant of origin-centered circle of diameter n.

Original entry on oeis.org

0, 0, 1, 1, 3, 4, 6, 8, 13, 15, 19, 22, 28, 30, 37, 41, 48, 54, 64, 69, 77, 83, 94, 98, 110, 119, 131, 139, 152, 162, 172, 183, 199, 208, 226, 234, 253, 263, 281, 294, 308, 322, 343, 357, 377, 390, 412, 424, 447, 465, 488, 504, 528, 545, 567, 585, 612, 628, 654

Offset: 1

Glenn C. Foster (gfoster(AT)uiuc.edu), Jan 02 2008

			a(5) = 3 because a circle of radius 5/2 in the first quadrant encloses (2,1), (1,1), (1,2).

Ivan Panchenko, Table of n, a(n) for n = 1..1000

Alternating merge of A136484 and A001182.

Cf. A019683, A136485, A136513.

A136483:= func< n | n eq 1 select 0 else (&+[Floor(Sqrt((n/2)^2-j^2)): j in [1..Floor(n/2)]]) >;
[A136483(n): n in [1..100]]; // G. C. Greubel, Jul 28 2023

Table[Sum[Floor[Sqrt[(n/2)^2 -k^2]], {k,Floor[n/2]}], {n,100}]

a(n) = sum(k=1, n\2, sqrtint((n/2)^2 - k^2)); \\ Michel Marcus, Jul 28 2023

def A136483(n): return sum(isqrt((n/2)^2-j^2) for j in range(1,(n//2)+1))
[A136483(n) for n in range(1,101)] # G. C. Greubel, Jul 28 2023

a(n) = Sum_{k=1..floor(n/2)} floor(sqrt((n/2)^2 - k^2)).
Lim_{n -> oo} a(n)/(n^2) -> Pi/16 (A019683).
a(n) = (1/4) * A136485(n) = (1/2) * A136513(n).
a(n) = [x^(n^2)] (theta_3(x^4) - 1)^2 / (4 * (1 - x)). - Ilya Gutkovskiy, Nov 23 2021

A136484 Number of unit square lattice cells inside quadrant of origin centered circle of diameter 2n+1.

Original entry on oeis.org

0, 1, 3, 6, 13, 19, 28, 37, 48, 64, 77, 94, 110, 131, 152, 172, 199, 226, 253, 281, 308, 343, 377, 412, 447, 488, 528, 567, 612, 654, 703, 750, 796, 847, 902, 957, 1013, 1068, 1129, 1187, 1252, 1313, 1378, 1446, 1511, 1582, 1650, 1725, 1800, 1877, 1955, 2034

Offset: 0

Glenn C. Foster (gfoster(AT)uiuc.edu), Jan 02 2008

Number of unit square lattice cells inside quadrant of origin centered circle of radius n+1/2.

			a(2) = 3 because a circle of radius 2+1/2 in the first quadrant encloses (2,1), (1,1), (1,2).

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A019683, A136483, A136486, A136515.

A136484:= func< n | n eq 0 select 0 else (&+[Floor(Sqrt((n+1/2)^2-j^2)): j in [1..n]]) >;
[A136484(n): n in [0..100]]; // G. C. Greubel, Jul 29 2023

Table[Sum[Floor[Sqrt[(n+1/2)^2 - k^2]], {k,n}], {n,0,100}]

def A136484(n): return sum(floor(sqrt((n+1/2)^2-k^2)) for k in range(1, n+1))
[A136484(n) for n in range(101)] # G. C. Greubel, Jul 29 2023

a(n) = Sum_{k=1..n} floor(sqrt((n+1/2)^2 - k^2)).
a(n) = (1/2) * A136515(n).
a(n) = (1/4) * A136486(n).
a(n) = A136483(2*n+1).
Lim_{n -> oo} a(n)/(n^2) -> Pi/16 (A019683).

A157332 Denominators of Egyptian fraction for Pi/16 based on Machin's formula.

Original entry on oeis.org

5, -956, -375, 163823028, 15625, -15596225303980, -546875, 1247220779824098212, 17578125, -91597497639855832244124, -537109375, 6394838587727583881086964116, 15869140625, -431694043145875922302762745864588, -457763671875

Offset: 0

Jaume Oliver Lafont, Feb 27 2009

Machin's formula: Pi/4 = 4*atan(1/5) - atan(1/239).

Sum_{n>=0} 1/a(n) = Pi/16 = atan(1/5) - (1/4)*atan(1/239).

G. C. Greubel, Table of n, a(n) for n = 0..415
X. Gourdon and P. Sebah, Collection of series for Pi

Cf. A019683, A072172, A072173.

a:= function(n)
    if n mod 2=0 then return (-1)^(n/2)*(n+1)*5^(n+1);
    else return -4*(-1)^((n-1)/2)*n*(239)^n;
    fi;
  end;
List([0..15], n-> a(n) ); # G. C. Greubel, Aug 26 2019

R:=PowerSeriesRing(Integers(), 15); Coefficients(R!( 5*(1-(5*x)^2)/(1+(5*x)^2)^2 - 4*239*x*(1-(239*x)^2)/(1+(239*x)^2)^2 )); // G. C. Greubel, Aug 26 2019

seq(coeff(series(5*(1-(5*x)^2)/(1+(5*x)^2)^2 - 4*239*x*(1-(239*x)^2)/(1+(239*x)^2)^2, x, n+1), x, n), n = 0..15); # G. C. Greubel, Aug 26 2019

CoefficientList[Series[5*(1-(5*x)^2)/(1+(5*x)^2)^2 - 4*239*x*(1-(239*x)^2)/(1+(239*x)^2)^2, {x,0,15}], x] (* G. C. Greubel, Aug 26 2019 *)

my(x='x+O('x^15)); Vec(5*(1-(5*x)^2)/(1+(5*x)^2)^2 - 4*239*x*(1-(239*x)^2)/(1+(239*x)^2)^2) \\ G. C. Greubel, Aug 26 2019

def A077952_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 5*(1-(5*x)^2)/(1+(5*x)^2)^2 - 4*239*x*(1-(239*x)^2)/(1+(239*x)^2)^2 ).list()
A077952_list(15) # G. C. Greubel, Aug 26 2019

a(2n) = (2*n+1)*5^(2*n+1)*(-1)^n,
a(2n+1) = -4*(2*n+1)*239^(2*n+1)*(-1)^n.
G.f.: 5*(1-25*x^2)/(1+25*x^2)^2 - 956*x*(1-57121*x^2)/(1+57121*x^2)^2

More terms from Colin Barker, Aug 07 2013

A019706 Decimal expansion of sqrt(Pi)/4.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

			0.44311346272637900682454187083528629569938736403059678...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
I. S. Gradsteyn and I. M. Ryzhik, Table of integrals, series and products, (1980), page 420 (formulas 3.757.1, 3.757.2).
Index entries for transcendental numbers

Cf. A019683.

evalf(sqrt(Pi)/4,120); # Muniru A Asiru, Sep 22 2018

First[RealDigits[Sqrt[Pi]/4, 10, 100]] (* Paolo Xausa, May 02 2024 *)

PARI
```
sqrt(Pi)/4 \\ Altug Alkan, Sep 22 2018
```

intnum(x=0, [oo, -8*I], sin(8*x)/sqrt(x)) \\ Gheorghe Coserea, Sep 23 2018

intnum(x=[0, -1/2], [oo, 8*I], cos(8*x)/sqrt(x)) \\ Gheorghe Coserea, Sep 23 2018

Equals sqrt(A019683). - Michel Marcus, Aug 31 2014
From A.H.M. Smeets, Sep 22 2018: (Start)
Equals Integral_{x >= 0} sin(8x)/sqrt(x) dx [Gradshteyn and Ryzhik].
Equals Integral_{x >= 0} cos(8x)/sqrt(x) dx [Gradshteyn and Ryzhik]. (End)
From Amiram Eldar, Aug 13 2020: (Start)
Equals Integral_{x=0..oo} x * exp(-x^4) dx.
Equals Integral_{x=0..oo} x^2 * exp(-x^2) dx. (End)

A069071 a(n) = (2n + 1)((2*n + 1)^4 + 4).

Original entry on oeis.org

5, 255, 3145, 16835, 59085, 161095, 371345, 759435, 1419925, 2476175, 4084185, 6436435, 9765725, 14349015, 20511265, 28629275, 39135525, 52522015, 69344105, 90224355, 115856365, 147008615, 184528305, 229345195, 282475445, 345025455, 418195705, 503284595, 601692285

Offset: 0

Benoit Cloitre, Apr 05 2002

The formula for Pi in the formula section was discovered by the mathematician and astronomer Nilakantha Somayaji (1444-1544) (Roy, 1990). - Amiram Eldar, Jan 18 2023

Amiram Eldar, Table of n, a(n) for n = 0..10000
Ranjan Roy, The Discovery of the Series Formula for Pi by Leibniz, Gregory and Nilakantha, Mathematics Magazine, Vol. 63, No. 5 (Dec., 1990), pp. 291-306.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A019683.

a[n_] := (2*n + 1)*((2*n + 1)^4 + 4); Array[a, 30, 0] (* Amiram Eldar, Jul 16 2022 *)

Pi = 16 * Sum_{n>=0} (-1)^n/a(n).
From Elmo R. Oliveira, Sep 03 2025: (Start)
G.f.: 5*(1 + x)*(1 + 44*x + 294*x^2 + 44*x^3 + x^4)/(x-1)^6.
E.g.f.: (5 + 250*x + 1320*x^2 + 1360*x^3 + 400*x^4 + 32*x^5)*exp(x).
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6). (End)

A132706 Decimal expansion of 16/Pi.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 31 2007

			5.092958178940650744604280427920459585102708663694606359925355....

Bruce C. Berndt, Ramanujan’s Notebooks, Part II, Springer-Verlag, New York, 1989.

Index entries for transcendental numbers

Cf. A019683, A049541, A060294, A089491, A088538, A086201, A132696 to A132699, A132701, A132702.

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

16/Pi \\ Charles R Greathouse IV, Oct 01 2022

Equals 4 + Sum_{k>=0} binomial(2*k,k)^2/((k+1)^2*16^k). - Amiram Eldar, May 21 2021

16/Pi = 5 + 1^2/(10 + 3^2/(10 + 5^2/(10 + ...))). See Berndt, Entry 25, p. 140, with n = 0 and x = 5. - Peter Bala, Feb 18 2024

More terms from Vladimir Joseph Stephan Orlovsky, Dec 02 2009

cp's OEIS Frontend

A017113 a(n) = 8*n + 4.

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A019679 Decimal expansion of Pi/12.

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A244978 Decimal expansion of Pi/32.

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A136483 Number of unit square lattice cells inside quadrant of origin-centered circle of diameter n.

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A136484 Number of unit square lattice cells inside quadrant of origin centered circle of diameter 2n+1.

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A157332 Denominators of Egyptian fraction for Pi/16 based on Machin's formula.

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A069071 a(n) = (2n + 1)((2*n + 1)^4 + 4).