OFFSET
0,2
COMMENTS
a(n) is the sum of the numerator and denominator of (n+1)/(2*n) + (n+2)/(2*(n+1)); all fractions are reduced and n > 0. - J. M. Bergot, Jun 14 2017
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
a(n) = a(n-1) + 8*n + 2. - Vincenzo Librandi, Aug 08 2010
From Michael De Vlieger, Jun 15 2017: (Start)
G.f.: (1 + 8*x - x^2)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
Sum_{k=1..n} a(k-1)/(2*k)! = 1 - 1/(2*n)!. - Robert Israel, Jul 19 2017
E.g.f.: (1 + 10*x + 4*x^2)*exp(x). - G. C. Greubel, Dec 22 2022
MATHEMATICA
(* Programs from Michael De Vlieger, Jun 15 2017 *)
Table[4n^2 +6n +1, {n, 0, 50}]
LinearRecurrence[{3, -3, 1}, {1, 11, 29}, 51]
CoefficientList[Series[(1+8*x-x^2)/(1-x)^3, {x, 0, 50}], x] (* End *)
PROG
(PARI) a(n)=4*n^2+6*n+1 \\ Charles R Greathouse IV, Oct 07 2015
(Magma) [4*n^2+6*n+1: n in [0..60]]; // G. C. Greubel, Dec 22 2022
(SageMath) [4*n^2+6*n+1 for n in range(61)] # G. C. Greubel, Dec 22 2022
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Apr 03 2003
EXTENSIONS
Incorrect formula and useless examples deleted by R. J. Mathar, Aug 31 2010
STATUS
approved