A035328 a(n) = n*(2*n-1)*(2*n+1).

0, 3, 30, 105, 252, 495, 858, 1365, 2040, 2907, 3990, 5313, 6900, 8775, 10962, 13485, 16368, 19635, 23310, 27417, 31980, 37023, 42570, 48645, 55272, 62475, 70278, 78705, 87780, 97527, 107970, 119133, 131040, 143715, 157182, 171465, 186588

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Bisection of A027480. For n>1, gives area of triangle two of whose cevians bound three smaller triangles with areas n-1, n, n+1 contiguously. - Lekraj Beedassy, Dec 21 2006

References

• Eric Harold Neville, Jacobian Elliptic Functions, 2nd ed., 1951, p. 38.
• Konrad Knopp, Theory and Application of Infinite Series, Dover, p. 269

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Konrad Knopp, Theorie und Anwendung der unendlichen Reihen, Berlin, J. Springer, 1922. (Original German edition of "Theory and Application of Infinite Series")

Crossrefs

Cf. A000292, A000447, A027480, A046092, A204558, A244009.

Programs

Magma

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

Mathematica

Table[n(2n-1)(2n+1),{n,0,40}] (* Harvey P. Dale, Jan 11 2014 *)

PARI

vector(100,n,(n-1)*(2*n-1)*(2*n-3)) \\ Derek Orr, Jan 29 2015

Formula

a(n) = 3*A000447(n) = 3*A000292(2*n-1).

Sum_{n>=1} 1/a(n) = 2*log(2) - 1. - Benoit Cloitre, Apr 05 2002

a(n) = A204558(2*n) / (2*n). - Reinhard Zumkeller, Jan 18 2012

G.f.: 3*x*(1 + 6*x + x^2)/(1 - x)^4. - Colin Barker, Mar 27 2012

Product_{n>=1} 4*n^3/a(n) = Pi/2. - Daniel Suteu, Feb 05 2017

a(n) = Sum_{i=0..2*n} A046092(n-1)+i = Sum_{i=2*n+1..4*n-1} A046092(n-1)+i for n>0. Example: for n = 5, A046092(4) = 40 and a(5) = 40 + 41 + 42 + ... + 49 + 50 = 51 + 52 + 53 + ... + 58 + 59 = 495. - Bruno Berselli, Oct 26 2017

Sum_{n>=1} (-1)^(n+1)/a(n) = 1 - log(2) (A244009). - Amiram Eldar, Jan 30 2021

E.g.f.: exp(x)*x*(3 + 12*x + 4*x^2). - Stefano Spezia, Sep 03 2023

Extensions

More terms from Benoit Cloitre, Apr 05 2002