OFFSET
0,2
COMMENTS
a(n) is the area of an irregular quadrilateral with vertices at (1,1), (n+1, n+2), ((n+1)^2, (n+2)^2) and ((n+1)^3, (n+2)^3). - Art Baker, Dec 08 2018
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..5000 (terms 0..680 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
FORMULA
G.f.: 12*x/(1 - x)^5. - Colin Barker, Mar 01 2012
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) with a(0) = 0, a(1) = 12, a(2) = 60, a(3) = 180, a(4) = 420. - Harvey P. Dale, Feb 04 2015
E.g.f.: (24*x + 36*x^2 + 12*x^3 + x^4)*exp(x)/2. - Franck Maminirina Ramaharo, Dec 08 2018
From Amiram Eldar, Sep 04 2022: (Start)
Sum_{n>=1} 1/a(n) = 1/9.
Sum_{n>=1} (-1)^(n+1)/a(n) = 8*(3*log(2)-2)/9. (End)
MAPLE
[seq(12*binomial(n+3, 4), n=0..32)]; # Zerinvary Lajos, Nov 24 2006
MATHEMATICA
Table[n*(n + 1)*(n + 2)*(n + 3)/2, {n, 0, 50}] (* David Nacin, Mar 01 2012 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 12, 60, 180, 420}, 40] (* Harvey P. Dale, Feb 04 2015 *)
PROG
(Magma) [n*(n+1)*(n+2)*(n+3)/2: n in [0..40]]; // Vincenzo Librandi, Apr 28 2011
(PARI) a(n)=n*(n+1)*(n+2)*(n+3)/2 \\ Charles R Greathouse IV, Oct 07 2015
(GAP) List([0..40], n->n*(n+1)*(n+2)*(n+3)/2); # Muniru A Asiru, Dec 08 2018
(Sage) [12*binomial(n+3, 4) for n in range(40)] # G. C. Greubel, Dec 08 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved