A244977 -id:A244977 - cp's OEIS frontend

A244978 Decimal expansion of Pi/32.

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)

A022778 Place where n-th 1 occurs in A023116.

Original entry on oeis.org

1, 2, 4, 6, 9, 12, 16, 21, 26, 32, 38, 45, 52, 60, 69, 78, 88, 98, 109, 120, 132, 145, 158, 172, 186, 201, 217, 233, 250, 267, 285, 303, 322, 342, 362, 383, 404, 426, 448, 471, 495, 519, 544, 569, 595, 621, 648, 676, 704, 733, 762, 792, 823, 854

Offset: 1

Clark Kimberling

nonn

Also place where first n appears in A007337. - Franklin T. Adams-Watters, Nov 10 2006

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A022776, A244977.

Table[n + 1 + Sum[Floor[(n - k)/Sqrt[3]], {k, 0, n}], {n, 0, 200}] (* A022778 *)
(* Clark Kimberling, Mar 14 2015 *)

a(n) = n + 1 + Sum{floor[(n - k)/sqrt(3)], k = 0..n}.

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

A144410 a(n) = 4(3n+1)(3n+2).

Original entry on oeis.org

8, 80, 224, 440, 728, 1088, 1520, 2024, 2600, 3248, 3968, 4760, 5624, 6560, 7568, 8648, 9800, 11024, 12320, 13688, 15128, 16640, 18224, 19880, 21608, 23408, 25280, 27224, 29240, 31328, 33488, 35720, 38024, 40400, 42848, 45368, 47960, 50624, 53360, 56168, 59048, 62000, 65024, 68120, 71288, 74528, 77840, 81224, 84680, 88208, 91808, 95480

Offset: 0

Paul Curtz, Sep 30 2008

The sequence lists all numbers k such that k+1 is a square and k+4 is divisible by 12. - Bruno Berselli, Sep 28 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A060544, A136016, A244977.

[4*(3*n+1)*(3*n+2): n in [0..40]]; // Vincenzo Librandi, Aug 07 2011

A144410:= n-> 4*(3*n+1)*(3*n+2); seq(A144410(n), n=0..60); # G. C. Greubel, Mar 27 2021

Table[4 (3 n + 1) (3 n + 2), {n, 0, 51}] (* or *)
CoefficientList[Series[8 (1 + 7 x + x^2)/(1 - x)^3, {x, 0, 51}], x] (* Michael De Vlieger, Sep 29 2017 *)

a(n)=4*(3*n+1)*(3*n+2) \\ Charles R Greathouse IV, Jun 17 2017

[4*(3*n+1)*(3*n+2) for n in (0..60)] # G. C. Greubel, Mar 27 2021

G.f.: 8*(1 + 7*x + x^2)/(1 - x)^3. - Michael De Vlieger, Sep 29 2017
a(n) = 8*A060544(n+1).
a(n) = A136016(2*n+1).
a(n) = a(m) + 36*(n - m)*(n + m + 1). For m = n-1, a(n) = a(n-1) + 72*n. - Bruno Berselli, Sep 29 2017
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), n >= 3. - Klaus Purath, Jul 05 2020
E.g.f.: 4*(2 +18*x +9*x^2)*exp(x). - G. C. Greubel, Mar 27 2021
From Amiram Eldar, Dec 10 2022: (Start)
Sum_{n>=0} 1/a(n) = Pi/(12*sqrt(3)) (A244977).
Sum_{n>=0} (-1)^n/a(n) = log(2)/6. (End)

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A244978 Decimal expansion of Pi/32.

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A022778 Place where n-th 1 occurs in A023116.

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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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A144410 a(n) = 4*(3*n+1)*(3*n+2).

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A144410 a(n) = 4(3n+1)(3n+2).