OFFSET
0,2
COMMENTS
LINKS
Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
FORMULA
O.g.f.: 3*x*(2 - x)/(1 - x)^4.
E.g.f.: x*(12 + 9*x + x^2)*exp(x)/2.
a(n) = n*(n + 1)*(n + 5)/2.
a(n) = Sum_{i=0..n} n*(n - i) + 5*i, that is: a(n) = A002411(n) + A028895(n). More generally, Sum_{i=0..n} n*(n - i) + k*i = n*(n + 1)*(n + k)/2.
a(n) = 3*A005581(n+1).
a(n+1) - 3*a(n) + 3*a(n-1) = 3*A105163(n) for n>0.
From Amiram Eldar, Jan 06 2021: (Start)
Sum_{n>=1} 1/a(n) = 163/600.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/5 - 253/600. (End)
EXAMPLE
The sequence is also provided by the row sums of the following triangle (see the fourth formula above):
. 0;
. 1, 5;
. 4, 7, 10;
. 9, 11, 13, 15;
. 16, 17, 18, 19, 20;
. 25, 25, 25, 25, 25, 25;
. 36, 35, 34, 33, 32, 31, 30;
. 49, 47, 45, 43, 41, 39, 37, 35;
. 64, 61, 58, 55, 52, 49, 46, 43, 40;
. 81, 77, 73, 69, 65, 61, 57, 53, 49, 45, etc.
First column is A000290.
Second column is A027690.
Third column is included in A189834.
Diagonal 1, 11, 25, 43, 65, 91, 121, ... is A161532.
MATHEMATICA
Table[n (n + 1) (n + 5)/2, {n, 0, 50}]
LinearRecurrence[{4, -6, 4, -1}, {0, 6, 21, 48}, 50] (* Harvey P. Dale, Jul 18 2019 *)
PROG
(PARI) vector(50, n, n--; n*(n+1)*(n+5)/2)
(Sage) [n*(n+1)*(n+5)/2 for n in (0..50)]
(Magma) [n*(n+1)*(n+5)/2: n in [0..50]];
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Bruno Berselli, Jan 13 2016
STATUS
approved