A069072 a(n) = (2n+1)*(2n+2)*(2n+3).
6, 60, 210, 504, 990, 1716, 2730, 4080, 5814, 7980, 10626, 13800, 17550, 21924, 26970, 32736, 39270, 46620, 54834, 63960, 74046, 85140, 97290, 110544, 124950, 140556, 157410, 175560, 195054, 215940, 238266, 262080, 287430, 314364, 342930
Offset: 0
References
- T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 190.
- Jolley, Summation of Series, Oxford (1961).
- Konrad Knopp, Theory and application of infinite series, Dover, p. 269.
Links
- M. Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
- Konrad Knopp, Theorie und Anwendung der unendlichen Reihen, Berlin, J. Springer, 1922. (Original german edition of "Theory and Application of Infinite Series")
- S. Ramanujan, Notebook entry
- Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
Programs
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Mathematica
Array[Times@@(2#+{1,2,3})&,40,0] (* or *) LinearRecurrence[{4,-6,4,-1},{6,60,210,504},40] (* Harvey P. Dale, Dec 08 2013 *) -
PARI
a(n)=(2*n+1)*(2*n+2)*(2*n+3) \\ Charles R Greathouse IV, Oct 07 2015
Formula
log(2) - 1/2 = Sum_{n>=0} 1/a(n); (1/2)*(1-log(2)) = Sum_{n>=0} (-1)^n/a(n). [Jolley eq 236 and 237]
Sum_{n>=0} x^n/a(n) = ((1+x)/sqrt(x)*log((1+sqrt x)/(1-sqrt x)) + 2*log(1-x)-2)/(4x). [Jolley eq 280 for 0
Sum_{n>=0} (-x)^n/a(n) = (1-log(1+x) -(1-x)/sqrt(x)*arctan(x))/(2x). [Jolley eq 281 for 0
a(n) = 6*A000447(n+1). - Lekraj Beedassy, Apr 18 2003
G.f.: 6*(1 + 6*x + x^2) / (x-1)^4 . - R. J. Mathar, Jun 09 2013
a(0)=6, a(1)=60, a(2)=210, a(3)=504, a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Harvey P. Dale, Dec 08 2013
a(n) = 2*A035328(n+1). - J. M. Bergot, Jan 02 2015
A097321 a(n) = (3*n-1) * 3*n * (3*n+1).
24, 210, 720, 1716, 3360, 5814, 9240, 13800, 19656, 26970, 35904, 46620, 59280, 74046, 91080, 110544, 132600, 157410, 185136, 215940, 249984, 287430, 328440, 373176, 421800, 474474, 531360, 592620, 658416, 728910, 804264, 884640, 970200, 1061106, 1157520
Offset: 1
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..10000
- S. Ramanujan, Notebook entry.
- Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
Programs
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Magma
[27*n^3-3*n: n in [1..40]]; // Vincenzo Librandi, Sep 07 2011
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Mathematica
Table[27n^3-3n,{n,40}] (* Harvey P. Dale, Mar 30 2011 *)
Formula
G.f.: 6x * (4x^2 + 19x + 4) / (1-x)^4.
Sum_{n>=1} 1/a(n) = (log(3) - 1)/2. - Amiram Eldar, Jul 04 2020
Sum_{n>=1} (-1)^(n+1)/a(n) = 1/2 - 2*log(2)/3. - Amiram Eldar, May 15 2022
E.g.f.: 3*exp(x)*x*(8 + 27*x + 9*x^2). - Stefano Spezia, Feb 20 2025
A084660 Decimal expansion of solution of area bisectors problem.
0, 1, 9, 8, 6, 0, 3, 8, 5, 4, 1, 9, 9, 5, 8, 9, 8, 2, 0, 6, 2, 9, 2, 4, 0, 9, 1, 0, 9, 3, 6, 3, 2, 4, 2, 6, 0, 5, 6, 6, 2, 5, 1, 0, 0, 7, 7, 0, 1, 9, 1, 4, 4, 0, 5, 9, 0, 5, 1, 0, 0, 0, 7, 1, 2, 0, 0, 4, 5, 2, 1, 6, 4, 7, 7, 2, 7, 1, 0, 3, 6, 7, 0, 4, 3, 9, 7, 4, 9, 5, 2, 4, 7, 3, 1, 4, 0, 1, 5, 6, 5, 6, 5
Offset: 0
Examples
0.0198603854199589820629240910936324260566251...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..10000
- Henry Bottomley, Area bisectors of a triangle.
- Zak Seidov 3-points in 3-4-5 triangle
Programs
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Magma
SetDefaultRealField(RealField(119)); Log(8/Exp(2))/4 // G. C. Greubel, Mar 22 2023
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Mathematica
Join[{0}, RealDigits[N[3/4*Log[2]-1/2, 108]][[1]]] (* Georg Fischer, Jul 15 2021 *) -
PARI
3*log(2)/4-1/2 \\ Charles R Greathouse IV, Apr 13 2020
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SageMath
numerical_approx(log(8/exp(2))/4, digits=119) # G. C. Greubel, Mar 22 2023
Formula
Equals (3*log(2) - 2)/4.
Sum_{i>0} 1/((4i-1)*4i*(4i+1)) = Sum_{i>0} 1/A069140(i). - Henry Bottomley, Jul 09 2003
Extensions
a(100) corrected by Georg Fischer, Jul 15 2021
Comments