A249649 -id:A249649 - cp's OEIS frontend

A179824 Chromatic polynomial of the star graph on 4 vertices (claw graph) and the path graph on 4 vertices.

2, 24, 108, 320, 750, 1512, 2744, 4608, 7290, 11000, 15972, 22464, 30758, 41160, 54000, 69632, 88434, 110808, 137180, 168000, 203742, 244904, 292008, 345600, 406250, 474552, 551124, 636608, 731670, 837000, 953312, 1081344, 1221858

Offset: 2

Jaime Soffer (jaime.soffer(AT)gmail.com), Jul 28 2010

To generate a Pythagorean triangle one uses (a,b) to get sides b^2-a^2, 2*a*b, and a^2+b^2, having a perimeter of 2*b*(a+b). If for a one uses the triangular number n*(n+1)/2 and for b the next triangular number (n+1)*(n+2)/2, the perimeter of the triangle so formed is (n+1)^3 * (n+2), which will give the same results as this sequence starting at the second term. - J. M. Bergot, Apr 01 2012
Define b(0)=0 and b(n)=A179824(n+1) for n > 0. Then b(n) is the number of 4-tuples (w,x,y,z) having all terms in {0,...,n} and no two consecutive terms equal. - Clark Kimberling, May 31 2012
Let n points in the plane each become the centers of n-1 concentric circles, circles that pass through only one of each of the other points. The maximum number of intersections of these circles is this sequence. [The solution was given by Andrew Weimholt in the Sequence Fans Mailing List] - J. M. Bergot, Mar 10 2014
Both the 'claw graph', a graph with 4 vertices where one vertex is adjacent to the other three, and the path graph on 4 vertices (per Clark Kimberling's comment), have this sequence as their chromatic polynomial, or the number of proper colorings of the graph using at most n colors. This is the standard example of two graphs which are not isomorphic, but which have the same chromatic polynomial.

			From _Jack W Grahl_, Jul 16 2018: (Start)
Consider the claw graph, which has vertices A, B, C, D, and edges AB, AC, AD. To color this graph with 3 colors, we can choose any of the 3 colors for A. Then each of the other vertices can be colored with any of the remaining two colors, giving 3 * 2 * 2 * 2 = 24 choices in all.
Similarly, consider the path graph with the same vertices and edges AB, BC, CD. We have 3 choices for the color of A, then 2 choices for the color of B (any color except that chosen for A), 2 choices for the color of C (any color except B's) etc. (End)

Vincenzo Librandi, Table of n, a(n) for n = 2..1000
Andrew Weimholt, Re: Intersecting circles, SeqFan post Mar 09 2014.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A019582, A179824.

List([2..40], n-> n*(n-1)^3 ); # G. C. Greubel, Aug 10 2019

let f n = [ (x,a,b,c) | let t = [1..n], x <- t, a <- t, x /= a, b <- t, x /= b, c <- t, x /= c ] in map (length.f) [2..]

Haskell
```
let f n = n*(n-1)^3 in map f [2..]
```

I:=[2,24,108,320,750]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Mar 12 2014

CoefficientList[Series[2 (1 + 7 x + 4 x^2)/(1 - x)^5, {x, 0, 50}], x] (* Vincenzo Librandi, Mar 12 2014 *)
Table[n^3+n^4,{n,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{2,24, 108,320,750},40] (* Harvey P. Dale, Sep 05 2015 *)

a(n) = n*(n-1)^3 \\ Charles R Greathouse IV, Mar 11 2014

[n*(n-1)^3 for n in (2..40)] # G. C. Greubel, Aug 10 2019

a(n) = n*(n-1)^3. - Jaime Soffer (jaime.soffer(AT)gmail.com), Jul 30 2010
G.f.: 2*x^2*(1 + 7*x + 4*x^2)/(1-x)^5. - Colin Barker, Jan 30 2012
a(n) = 2*A019582(n). - R. J. Mathar, Jun 09 2013
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Vincenzo Librandi, Mar 12 2014
Sum_{n>=2} 1/a(n) = A249649. - R. J. Mathar, Oct 18 2019
Sum_{n>=2} (-1)^n/a(n) = 3*zeta(3)/4 + 2*log(2) - Pi^2/12 - 1. - Amiram Eldar, Nov 05 2020

Name edited by Jack W Grahl, Jul 16 2018

A249651 Decimal expansion of Integral_{0..1} Li_2(x)^2 dx, where Li_2 is the dilogarithm function.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Nov 03 2014

			0.607712337943015464246226262015069415439032408...

Eric Weisstein's MathWorld, Dilogarithm

Cf. A002117, A013661, A249649.

RealDigits[6 - 2*Zeta[2] - 4*Zeta[3] + Zeta[2]^2, 10, 106] // First
NIntegrate[PolyLog[2,x]^2,{x,0,1},WorkingPrecision->110] (* Vaclav Kotesovec, Nov 03 2014 *)

from mpmath import *
mp.dps=107
f=lambda x: polylog(2, x)**2
I=quad(f, [0, 1])
print([int(n) for n in list(str(I)[2:-1])]) # Indranil Ghosh, Jul 04 2017

6 - 2*zeta(2) - 4*zeta(3) + zeta(2)^2.

A249652 Decimal expansion of integral_{0..1} Li_3(x)^2 dx, where Li_3 is the trilogarithm function.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Nov 03 2014

			0.4277147842908240881128389716127945324286...

Eric Weisstein's MathWorld, Trilogarithm

Cf. A002117, A013661, A013662, A249649, A249651.

RealDigits[20 - 8*Zeta[2] - 10*Zeta[3] + (15/2)*Zeta[4] - 2*Zeta[2]*Zeta[3] + Zeta[3]^2, 10, 102] // First
NIntegrate[PolyLog[3,x]^2,{x,0,1},WorkingPrecision->102] (* Vaclav Kotesovec, Nov 03 2014 *)

z2=zeta(2); z3=zeta(3); 20 - 8*z2 - 10*z3 + 15*zeta(4)/2 - 2*z2*z3 + z3^2 \\ Charles R Greathouse IV, Apr 20 2016

from mpmath import mp, zeta
mp.dps=103
z2=zeta(2)
z3=zeta(3)
print([int(z) for z in list(str(20 - 8*z2 - 10*z3 + 15*zeta(4)/2 - 2*z2*z3 + z3**2)[2:-1])]) # Indranil Ghosh, Jul 03 2017

20 - 8*zeta(2) - 10*zeta(3) + (15/2)*zeta(4) - 2*zeta(2)*zeta(3) + zeta(3)^2.

A365797 Decimal expansion of smallest positive number x such that Gamma(x) = 2.

Original entry on oeis.org

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

Offset: 0

David Ulgenes, Sep 19 2023

Second branch (i.e., the first after the principal branch) of the inverse gamma function Gamma(y) = x at x=2. See for instance Uchiyama.
Since 1 - x = 0.55712260351... (approximately equal to A249649), we can obtain the interesting approximation Gamma(zeta(2) - zeta(3)) ≈ 2.000001... - David Ulgenes, Feb 19 2024
x is the least positive real number where 1+Gamma(1+Gamma(1+Gamma...(x)...)) converges; it converges to 3. - Colin Linzer, Nov 25 2024

			0.4428773964847274374520325165206056717103645380663664...

K. Amenyo Folitse, David J. Jeffrey, and Robert M. Corless, Properties and Computation of the Functional Inverse of Gamma, 2017 19th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC). International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC). p. 65.
Mitsuru Uchiyama, The principal inverse of the gamma function, Proc. Amer. Math. Soc. 140 (2012), 1343-1348.

Cf. A030169, A030171.

Digits:= 140:
with(RootFinding):
NextZero(x -> (x - 1)! - 2, 0);

FindRoot[-2 + (-1 + x)! == 0, {x, 0, 1}, WorkingPrecision -> 15]

solve(x=0.1, 1, gamma(x)-2) \\ Michel Marcus, Sep 19 2023

Equals ((((1/2)!/2)!/2)!/2)!/2...
Proof: Since y = y! / x we substitute the expression into itself to obtain an iterative scheme for the inverse gamma function.
Equals (1/(2*Pi))*Integral_{x=-oo..oo} log((2-Gamma(i*x))/(2-Gamma(1+i*x))) dx. Proof: Follows from writing the inverse gamma function using the Lagrange inversion theorem together with Cauchy's formula for differentiation. - David Ulgenes, Feb 11 2024

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A179824 Chromatic polynomial of the star graph on 4 vertices (claw graph) and the path graph on 4 vertices.

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A249651 Decimal expansion of Integral_{0..1} Li_2(x)^2 dx, where Li_2 is the dilogarithm function.

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A249652 Decimal expansion of integral_{0..1} Li_3(x)^2 dx, where Li_3 is the trilogarithm function.

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A365797 Decimal expansion of smallest positive number x such that Gamma(x) = 2.

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