OFFSET
0,2
COMMENTS
Shares its digital root, zero together with period 9: repeat [3, 3, 3, 6, 6, 6, 9, 9, 9] with A027480.
Final digits cycle a length period 20: repeat [0, 9, 9, 2, 0, 5, 9, 4, 2, 5, 5, 4, 4, 7, 5, 0, 4, 9, 7, 0].
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
FORMULA
a(n) = 2*n^3 + 9*n^2/2 + 5*n/2.
a(n) = 3*A016061(n).
a(n) = A007742(n)*(n - 1)/2.
From Colin Barker, Jan 21 2017: (Start)
G.f.: 3*x*(3 + x) / (1 - x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3. (End)
From Stefano Spezia, Aug 30 2022: (Start)
E.g.f.: exp(x)*x*(18 + 21*x + 4*x^2)/2.
Sum_{n>0} 1/a(n) = 2*(20*log(8) + 10*Pi - 71)/25 = 0.1603805895595720759728288896228498341201... . (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*sqrt(2)*Pi/5 + 4*(3+sqrt(2))*log(2)/5 - 8*sqrt(2)*log(2-sqrt(2))/5 - 178/25. - Amiram Eldar, Sep 22 2022
MATHEMATICA
Table[n (n + 1) (4 n + 5)/2, {n, 0, 45}] (* or *)
CoefficientList[Series[3 x (3 + x)/(1 - x)^4, {x, 0, 45}], x] (* Michael De Vlieger, Jan 21 2017 *)
PROG
(PARI) concat(0, Vec(3*x*(3 + x) / (1 - x)^4 + O(x^50))) \\ Colin Barker, Jan 21 2017
(PARI) a(n) = n*(n + 1)*(4*n + 5)/2 \\ Charles R Greathouse IV, Feb 01 2017
(Magma) [n*(n+1)*(4*n+5)/2 : n in [0..50]]; // Wesley Ivan Hurt, Aug 30 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Peter M. Chema, Jan 21 2017
STATUS
approved