A060459 -id:A060459 - cp's OEIS frontend

A248619 a(n) = (n*(n+1))^4.

0, 16, 1296, 20736, 160000, 810000, 3111696, 9834496, 26873856, 65610000, 146410000, 303595776, 592240896, 1097199376, 1944810000, 3317760000, 5473632256, 8767700496, 13680577296, 20851360000, 31116960000, 45558341136, 65554433296, 92844527616, 129600000000

Offset: 0

Eugene Chong, Oct 09 2014

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A016744, A059977; A002378: n*(n+1); A035287: n^2 *(n-1)^2; A060459: n^3*(n+1)^3.

Cf. A327773.

[(n*(n+1))^4: n in [0..30]]; // Vincenzo Librandi, Oct 16 2014

Maple
```
[ seq(n^4*(n+1)^4, n = 0..100) ];
```

Table[(n (n + 1))^4, {n, 0, 70}] (* or *) CoefficientList[Series[16 x (1 + 72 x + 603 x^2 + 1168 x^3 + 603 x^4 + 72 x^5 + x^6)/(1 - x)^9, {x, 0, 30}], x] (* Vincenzo Librandi, Oct 16 2014 *)
LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{0,16,1296,20736,160000,810000,3111696,9834496,26873856},30] (* Harvey P. Dale, Sep 09 2016 *)

a(n) = A002378(n)^4 = A016744(A000217(n)).
a(n) = 16*A059977(n) for n>0.
G.f.: 16*x*(1 + 72*x + 603*x^2 + 1168*x^3 + 603*x^4 + 72*x^5 + x^6)/(1 - x)^9. - Vincenzo Librandi, Oct 16 2014
Sum_{n>=1} 1/a(n) = A327773 = -35 + 10*Pi^2/3 + Pi^4/45. - Vaclav Kotesovec, Sep 25 2019

Terms a(76) and beyond corrected by Andrew Howroyd, Feb 20 2018

A348670 Decimal expansion of 10 - Pi^2.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Oct 29 2021

Let ABC be a unit-area triangle, and let P be a point uniformly picked at random inside it. Let D, E and F be the intersection points of the lines AP, BP and CP with the sides BC, CA and AB, respectively. Then, the expected value of the area of the triangle DEF is this constant.

			0.13039559891064138116550900012384886468630059275920...

Calvin C. Clawson, Mathematical Mysteries: The Beauty and Magic of Numbers, Springer, 2013, p. 220.
A. M. Mathai, An introduction to geometrical probability: distributional aspects with applications, Amsterdam: Gordon and Breach, 1999, p. 275, ex. 2.5.3.

R. W. Gosper, Acceleration of Series, AIM-304 (1974), page 71.
Olivier Schneegans, How Close to 10 is Pi^2?, The American Mathematical Monthly, Vol. 126, No. 5 (2019), p. 448.
Daniel Sitaru, Problem B131, Crux Mathematicorum, Vol. 49, No. 7 (2023), p. 381.
Index entries for transcendental numbers.

Cf. A002388, A008911, A010467, A060459.

Mathematica
```
RealDigits[10 - Pi^2, 10, 100][[1]]
```
PARI
```
10 - Pi^2 \\ Michel Marcus, Oct 29 2021
```

Equals Sum_{k>=1} 1/(k*(k+1))^3 = Sum_{k>=1} 1/A060459(k).
Equals 6 * Sum_{k>=2} 1/(k*(k+1)^2*(k+2)) = Sum_{k>=3} 1/A008911(k).
Equals 2 * Integral_{x=0..1, y=0..1} x*(1-x)*y*(1-y)/(1-x*y)^2 dx dy.
Equals 4 * Sum_{m,n>=1} (m-n)^2/(m*n*(m+1)^2*(n+1)^2*(m+2)*(n+2)) (Sitaru, 2023). - Amiram Eldar, Aug 18 2023

A248720 a(n) = (n*(n+1))^5.

Original entry on oeis.org

0, 32, 7776, 248832, 3200000, 24300000, 130691232, 550731776, 1934917632, 5904900000, 16105100000, 40074642432, 92389579776, 199690286432, 408410100000, 796262400000, 1488827973632, 2682916351776, 4678757435232, 7923516800000, 13069123200000

Offset: 0

Eugene Chong, Oct 16 2014

This is the sequence (2^5)*A059860(n)= (2*binomial(n+1,2))^5, n >= 0. - Wolfdieter Lang, Nov 03 2014

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1).

Cf. A059860, A002378 (n*(n+1)), A035287(n+1) ((n*(n+1))^2), A060459 ((n*(n+1))^3), A248619 ((n*(n+1))^4).

Magma
```
[(n*(n+1))^5: n in [0..30]];
```
Maple
```
[ seq(n^5*(n+1)^5, n = 0..100) ];
```

Table[(n (n + 1))^5, {n, 0, 70}] (* or *) CoefficientList[Series[32 x (x^8 + 232 x^7 + 5158 x^6 + 27664 x^5 + 47290 x^4 + 27664 x^3 + 5158 x^2 + 232 x + 1)/(1 - x)^11, {x, 0, 30}], x]
LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{0,32,7776,248832,3200000,24300000,130691232,550731776,1934917632,5904900000,16105100000},20] (* Harvey P. Dale, Apr 23 2017 *)

a(n) = A002378(n)^5.
a(n) = 32*A059860(n) for n>0.
G.f.: 32*x*(x^8 + 232*x^7 + 5158*x^6 + 27664*x^5 + 47290*x^4 + 27664*x^3 + 5158*x^2 + 232*x + 1) / (1 - x)^11 (from A059860).
Sum_{n>=1} 1/a(n) = 126 - 35*Pi^2/3 - Pi^4/9. - Vaclav Kotesovec, Sep 25 2019
a(n) = 11*a(n-1) - 55*a(n-2) + 165*a(n-3) - 330*a(n-4) + 462*a(n-5) - 462*a(n-6) + 330*a(n-7) - 165*a(n-8) + 55*a(n-9) - 11*a(n-10) + a(n-11). - Wesley Ivan Hurt, Jan 20 2024

Terms a(32) and beyond corrected by Andrew Howroyd, Feb 20 2018

A249076 a(n) = (n*(n+1))^6.

Original entry on oeis.org

0, 64, 46656, 2985984, 64000000, 729000000, 5489031744, 30840979456, 139314069504, 531441000000, 1771561000000, 5289852801024, 14412774445056, 36343632130624, 85766121000000, 191102976000000, 404961208827904, 820972403643456, 1600135042849344, 3010936384000000, 5489031744000000

Offset: 0

Jiwoo Lee, Oct 20 2014

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A059978; A002378: n*(n+1); A035282: n^2 *(n+1)^2; A060459: n^3 *(n+1)^3; A248619: n^4 *(n+1)^4;

Magma
```
[(n*(n+1))^6: n in [0..30]];
```
Maple
```
[ seq(n^6*(n+1)^6, n = 0..100) ];
```

Table[(n (n + 1))^6, {n, 0, 70}] (* or *)
CoefficientList[Series[64*x*(x^10 + 716 x^9 + 37257 x^8 + 450048 x^7 + 1822014 x^6 + 2864328 x^5 + 1822014 x^4 + 450048 x^3 + 37257 x^2 + 716 x + 1)/(1 - x)^13, {x, 0, 30}], x]

a(n)=(n*(n+1))^6 \\ Charles R Greathouse IV, Oct 21 2014

a(n) = A002378(n)^6.
a(n) = 64*A059978(n) for n>0.
G.f.: 64*x*(x^10 + 716*x^9 + 37257*x^8 + 450048*x^7 + 1822014*x^6 + 2864328*x^5 + 1822014*x^4 + 450048*x^3 + 37257*x^2 + 716*x + 1)/(1 - x)^13. [corrected by Georg Fischer, May 10 2019]
Sum_{n>=1} 1/a(n) = -462 + 42*Pi^2 + 7*Pi^4/15 + 2*Pi^6/945. - Vaclav Kotesovec, Sep 25 2019

Incorrect term corrected by Colin Barker, Oct 21 2014

Terms a(21) and beyond corrected by Andrew Howroyd, Feb 22 2018

A377567 Decimal expansion of 3zeta(3)/2 + 12log(2) - 10.

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Nov 03 2024

			0.12085152145873514110639269976529380305348105083381

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

Michael I. Shamos, A catalog of the real numbers, (2007). See p. 159.

Cf. A002117, A002162, A060459.

RealDigits[3Zeta[3]/2+12Log[2]-10,10,100][[1]]

Equals Sum_{k>=1} (-1)^(k+1)/(k^3*(k + 1)^3) (see Finch and Shamos).

cp's OEIS Frontend

A248619 a(n) = (n*(n+1))^4.

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A348670 Decimal expansion of 10 - Pi^2.

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A248720 a(n) = (n*(n+1))^5.

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A249076 a(n) = (n*(n+1))^6.

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A377567 Decimal expansion of 3*zeta(3)/2 + 12*log(2) - 10.

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A377567 Decimal expansion of 3zeta(3)/2 + 12log(2) - 10.