A092425 -id:A092425 - cp's OEIS frontend

A164108 Decimal expansion of Pi^4/24.

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

Offset: 1

R. J. Mathar, Aug 10 2009

Volume of the 8-dimensional unit sphere.

			4.0587121264167682181850138620293796354053160696952259038...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Cornel Ioan Vălean, Problem 11993, The American Mathematical Monthly, Vol. 124, No. 7 (2017), p. 659; An Integral Related to Euler Sums, Solution to Problem 11993 by Abdelhak Berkane, ibid., Vol. 126, No. 4 (2019), pp. 373-374.
Eric Weisstein's World of Mathematics, Hypersphere.
Wikipedia, n-sphere.
Index entries for transcendental numbers

Cf. A000796, A019699, A102753, A164103, A164105, A164106.

RealDigits[Pi^4/24, 10, 100][[1]] (* G. C. Greubel, Apr 09 2017 *)

PARI
```
Pi^4/24 \\ G. C. Greubel, Apr 09 2017
```

Equals A164109/8 = A092425/24 = A072691*A102753.

Pi^4/240 = -Integral_{x=0..1} log(1-x)*log(1+x)^2/x dx (Vălean, 2017). - Amiram Eldar, Mar 26 2022

A276023 Decimal expansion of 32*Pi^4/945.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Aug 16 2016

Volume of the 9-dimensional unit sphere.

More generally, the ordinary generation function for the volume of the n-dimensional unit sphere is exp(Pi*x^2)*(erf(sqrt(Pi)*x) + 1) = 1 + 2*x + Pi*x^2 + (4*Pi/3)*x^3 + (Pi^2/2)*x^4 + ...

			3.2985089027387068693821065037445117...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Ilya Gutkovskiy, Volume of the n-dimensional unit sphere
Eric Weisstein's World of Mathematics, Hypersphere
Index entries for transcendental numbers

Cf. A021949, A092425.

Cf. similar sequences of the volume of the n-dimensional unit sphere: A000796 (n = 2), 10*A019699 (n = 3), A102753 (n = 4), A164103 (n = 5), A164105 (n = 6), A164106 (n = 7), A164108 (n = 8).

Mathematica
```
RealDigits[(32 Pi^4)/945, 10, 120][[1]]
```

(32*Pi^4)/945 \\ G. C. Greubel, Apr 09 2017

Equals 32*A092425*A021949.

A100322 a(n) is the smallest positive integer k such that the digits of the fractional part of Pi^k begin with n.

Original entry on oeis.org

1, 7, 6, 4, 8, 23, 25, 2, 15, 91, 51, 307, 49, 1, 102, 315, 112, 12, 76, 26, 115, 208, 77, 276, 161, 40, 13, 41, 7, 99, 174, 169, 86, 453, 110, 204, 53, 6, 67, 4, 228, 123, 37, 134, 158, 192, 33, 45, 61, 200, 31, 324, 8, 56, 34, 105, 148, 17, 19, 92, 23, 38, 27, 39, 32, 82

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Nov 16 2004

			Pi^1 = 3.14159..., whose digits after the decimal point begin with 1, so a(1)=1.
Pi^2 = 9.869..., whose digits after the decimal point begin with 8, so a(8)=2.
a(14)=1 because Pi^1 = 3.14....

Cf. A000796, A002388, A091925, A092425, A092731, A092732, A092735, A092736.

a(n) = my(k=1); while (floor(frac(Pi^k)*10^(1+logint(n, 10))) != n, k++); k; \\ Michel Marcus, Jun 18 2022

A193717 Decimal expansion of Pi^4log(2)/64 - 9Pi^2zeta(3)/64 + 93zeta(5)/128.

Original entry on oeis.org

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

Offset: 0

Seiichi Kirikami, Aug 03 2011

The absolute value of the integral {x=0..Pi/2} x^3*log(sin(x )) dx or (d^3/da^3 (integral {x=0..Pi/2} sin(ax )*log(sin(x )) dx)) at a=0. The absolute value of (sum {n=1..infinity} (limit { a -> 0} (d^3/da^3 ((1-cos((a+2n)*Pi/2))/n/(a+2n)))))-(Pi/2)^4*log(2)/4. [Seiichi Kirikami and Peter J. C. Moses]

			-0.14002410170685231710...

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, series and Products, 4th edition, 1.441.2, log(sin(x))=-(sum {1..infinity} cos(2nx)/n)-log(2).

S. Koyama and N. Kurokawa, Euler’s integrals and multiple sine functions, Proc. Amer. Math. Soc. 133(2005), 1257-1265.

Cf. A173623, A173624, A193716.

RealDigits[N[(2 Pi^4 Log[2] - 18 Pi^2 Zeta[3] + 93 Zeta[5]) / 128, 105]][[1]]

Pi^4*log(2)/64 - 9*Pi^2*zeta(3)/64 + 93*zeta(5)/128 \\ Michel Marcus, Oct 25 2017

Equals A092425*A002162/64-9*A002388*A002117/64+93*A013663/128.

A300731 Decimal expansion of sqrt(Pi^4/96 - 1).

Original entry on oeis.org

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

Offset: 0

Iaroslav V. Blagouchine, Mar 11 2018

Also the total harmonic distortion (THD) of a triangle wave, see formula (14) in the Blagouchine & Moreau link.

			0.1211529265193047433149738747453528509885975440568532...

I. V. Blagouchine and E. Moreau, Analytic Method for the Computation of the Total Harmonic Distortion by the Cauchy Method of Residues, IEEE Trans. Commun., vol. 59, no. 9, pp. 2478-2491, 2011. PDF file.
Index entries for transcendental numbers

Cf. A092425, A300690, A300707, A300713, A300714, A300727.

MATLAB
```
format long; sqrt(pi^4/96-1)
```
Maple
```
evalf(sqrt((1/96)*Pi^4-1), 120)
```

RealDigits[Sqrt[Pi^4/96 - 1], 10, 120][[1]]

default(realprecision, 120); sqrt(Pi^4/96-1)

A058286 Continued fraction for Pi^4.

Original entry on oeis.org

97, 2, 2, 3, 1, 16539, 1, 6, 7, 6, 8, 6, 3, 9, 1, 1, 1, 18, 1, 4, 1, 13, 1, 2, 1, 127, 1, 1, 1, 4, 1, 6, 1, 1, 1, 10, 10, 1, 1, 2, 1, 2, 1, 5, 1, 1, 10, 1, 3, 2, 1, 1, 4, 9, 1, 7, 70, 1, 13, 1, 2, 6, 1, 2, 24, 5, 2, 6, 1, 1, 1, 8, 1, 1, 11, 2, 1, 1, 4, 3, 1, 3, 2, 2, 1, 7, 1, 4, 1, 22, 2, 1, 2, 3, 1

Offset: 0

Robert G. Wilson v, Dec 07 2000

"Truncating just before the unexpectedly large partial quotient 16,539 gives a famous approximation of Ramanujan for Pi^4 of 97 9/22." (Wells)

			97.4090910340024372364403326... = 97 + 1/(2 + 1/(2 + 1/(3 + 1/(1 + ...)))). - _Harry J. Smith_, Jun 22 2009

David Wells, "The Penguin Dictionary of Curious and Interesting Numbers," Revised Edition, Penguin Books, London, England, 1997, page 116.

Harry J. Smith, Table of n, a(n) for n = 0..20000

Cf. A092425 Decimal expansion. - Harry J. Smith, Jun 22 2009

Mathematica
```
ContinuedFraction[ Pi^4, 100]
```

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(Pi^4); for (n=0, 20000, write("b058286.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 22 2009

A212004 Decimal expansion of (2*Pi)^4.

Original entry on oeis.org

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

Offset: 4

Omar E. Pol, Aug 11 2012

			1558.545456544038995783...

Juan M. Maldacena, The Large N Limit of Superconformal Field Theories and Supergravity, arXiv:hep-th/9711200v3, 1997-1998, p. 11.
Index entries for transcendental numbers

Cf. A000796, A019692, A092425, A212002, A212003, A212005.

RealDigits[(2*Pi)^4, 10, 100][[1]] (* Amiram Eldar, Jun 12 2021 *)

16*Pi^4 \\ Charles R Greathouse IV, Oct 01 2022

Equals Product_{k=1..15, gcd(k,15)==1} Gamma(k/15). - Amiram Eldar, Jun 12 2021

A269792 a(n) = 5*n^4.

Original entry on oeis.org

0, 5, 80, 405, 1280, 3125, 6480, 12005, 20480, 32805, 50000, 73205, 103680, 142805, 192080, 253125, 327680, 417605, 524880, 651605, 800000, 972405, 1171280, 1399205, 1658880, 1953125, 2284880, 2657205, 3073280, 3536405, 4050000, 4617605, 5242880, 5929605

Offset: 0

Ilya Gutkovskiy, Mar 31 2016

More generally, the ordinary generating function for the sequences of the form k*n^m, is k*Sum_{j>=1}x^j*j^m (when abs(x)<1).

More generally, the ordinary generating function for the values of quartic polynomial p*n^4 + q*n^3 + k*n^2 + m*n + r, is (r + (p + q + k + m - 4*r)*x + (11*p + 3*q - k - 3*m + 6*r)*x^2 + (11*p - 3*q - k + 3*m - 4*r)*x^3 + (p - q + k - m + r)*x^4)/(1 - x)^5.

Ilya Gutkovskiy, Examples of the ordinary generating function for the values of quartic polynomial
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1)

Cf. similar sequences of the form k*n^m, for k = 1...5, m = 1...10: A001477(k = 1, m = 1), A005843 (k = 2, m = 1), A008585 (k = 3, m = 1), A008586 (k = 4, m = 1), A008587 (k = 5, m = 1), A000290 (k = 1, m = 2), A001105 (k = 2, m = 2), A033428 (k = 3, m = 2), A016742 (k = 4, m = 2), A033429 (k = 5, m = 2), A000578 (k = 1, m = 3), A033431 (k = 2, m = 3), A117642 (k = 3, m = 3), A033430 (k = 4, m = 3), A244725 (k = 5, m = 3), A000583 (k = 1, m = 4), A244730 (k = 2, m = 4), A219056 (k = 3, m = 4), A141046 (k = 4, m = 4), this sequence(k = 5, m = 4), A000584 (k = 1, m = 5), A001014 (k = 1, m = 6), A106318 (k = 2, m = 6), A001015 (k = 1, m = 7), A001016 (k = 1, m = 8), A001017 (k = 1, m = 9), A008454 (k = 1, m = 10).

A269792:=n->5*n^4: seq(A269792(n), n=0..50); # Wesley Ivan Hurt, Apr 28 2017

Table[5 n^4, {n, 0, 33}]
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 5, 80, 405, 1280}, 34]

x='x+O('x^99); concat(0, Vec(5*x*(1+11*x+11*x^2+x^3)/(1-x)^5)) \\ Altug Alkan, Mar 31 2016

G.f.: 5*x*(1 + 11*x + 11*x^2 + x^3)/(1 - x)^5.
E.g.f.: 5*exp(x)^x*x*(1 + 7*x + 6*x^2 + x^3).
a(n) = 5*a(n-1) - 10*(9n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = 5*A000583(n) = A008587(n)*A000578(n).
Sum_{n>=1} 1/a(n) = Pi^4/450 = (1/450)*A092425 = 0.216464646742...

A306604 Number of perfect squares in the half-open interval [Pi^(n-1), Pi^n).

Original entry on oeis.org

0, 1, 2, 2, 4, 8, 14, 23, 43, 75, 134, 236, 419, 743, 1316, 2333, 4135, 7329, 12992, 23026, 40813, 72338, 128218, 227259, 402806, 713955, 1265453, 2242956, 3975538, 7046456, 12489518, 22137096, 39236979, 69545736, 123266607, 218484372, 387253468, 686388899

Offset: 0

Alois P. Heinz, Feb 27 2019

nonn

Inspired by A306486.

			a(4) = 4: in the interval [Pi^3, Pi^4) = [31.006..., 97.409...) = are four perfect squares: 36, 49, 64, 81.

Alois P. Heinz, Table of n, a(n) for n = 0..4024

Cf. A000290, A000796, A002161, A091925, A092425, A102475, A102477, A306486.

a:= n-> (f-> f(n)-f(n-1))(i-> ceil(Pi^(i/2))):
seq(a(n), n=0..42);

a(n) = ceiling(Pi^(n/2)) - ceiling(Pi^((n-1)/2)).
a(n) = A102477(n) - A102477(n-1).
Sum_{i=0..n} a(i) = A102475(n) for n > 0.
Lim_{n->oo} a(n+1)/a(n) = sqrt(Pi) = 1.7724538509... = A002161.

A377592 Decimal expansion of 9zeta(3)/Pi^2 - 93zeta(5)/(2*Pi^4) - log(2) + 1/4.

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Nov 02 2024

			0.158000963625557733268629385978458549091780284796...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, pp. 43-44.

Cf. A002117, A002162, A002388, A013663, A092425, A094642, A244009, A377588, A377589.

RealDigits[9Zeta[3]/Pi^2-93Zeta[5]/(2Pi^4)-Log[2]+1/4,10,100][[1]]

Equals Sum_{k>=1} zeta(2*k)/((k + 2)*2^(2*k)) (see Finch).

cp's OEIS Frontend

A164108 Decimal expansion of Pi^4/24.

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A276023 Decimal expansion of 32*Pi^4/945.

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A100322 a(n) is the smallest positive integer k such that the digits of the fractional part of Pi^k begin with n.

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A193717 Decimal expansion of Pi^4*log(2)/64 - 9*Pi^2*zeta(3)/64 + 93*zeta(5)/128.

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A300731 Decimal expansion of sqrt(Pi^4/96 - 1).

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A058286 Continued fraction for Pi^4.

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A212004 Decimal expansion of (2*Pi)^4.

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A193717 Decimal expansion of Pi^4log(2)/64 - 9Pi^2zeta(3)/64 + 93zeta(5)/128.

A377592 Decimal expansion of 9zeta(3)/Pi^2 - 93zeta(5)/(2*Pi^4) - log(2) + 1/4.