|
|
A084263
|
|
a(n) = (-1)^n/2+(n^2+n+1)/2.
|
|
8
|
|
|
1, 1, 4, 6, 11, 15, 22, 28, 37, 45, 56, 66, 79, 91, 106, 120, 137, 153, 172, 190, 211, 231, 254, 276, 301, 325, 352, 378, 407, 435, 466, 496, 529, 561, 596, 630, 667, 703, 742, 780, 821, 861, 904, 946, 991, 1035, 1082, 1128, 1177, 1225, 1276, 1326, 1379, 1431
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
COMMENTS
|
Old name was "Modified triangular numbers".
Starting with offset 1 = row sums of an infinite lower triangular matrix with alternate columns of (1, 3, 5, 7, ...) and (1, 0, 0, 0, ...) (see example). - Gary W. Adamson, May 14 2010
|
|
LINKS
|
|
|
FORMULA
|
E.g.f.: cosh(x)+exp(x)*(x+x^2/2).
a(n) = Sum_{k=0..n} k+(-1)^k.
G.f.: (1-x+2*x^2)/((1-x)^3*(1+x)). - R. J. Mathar, Apr 02 2008
|
|
EXAMPLE
|
First few rows of the triangle with row sums = A084263 =
1;
3, 1;
5, 0, 1;
7, 0, 3, 1;
9, 0, 5, 0, 1;
11, 0, 7, 0, 3, 1;
...
Example: a(4) = 11 = (7 + 0 + 3 + 1). (End)
|
|
MATHEMATICA
|
Table[(-1)^n/2 + (n^2 + n + 1)/2, {n, 0, 100}] (* Wesley Ivan Hurt, Dec 23 2021 *)
LinearRecurrence[{2, 0, -2, 1}, {1, 1, 4, 6}, 60] (* Harvey P. Dale, Jun 10 2023 *)
|
|
PROG
|
|
|
CROSSREFS
|
|
|
KEYWORD
|
easy,nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|