|
|
A228887
|
|
a(n) = binomial(3*n + 1,3).
|
|
4
|
|
|
4, 35, 120, 286, 560, 969, 1540, 2300, 3276, 4495, 5984, 7770, 9880, 12341, 15180, 18424, 22100, 26235, 30856, 35990, 41664, 47905, 54740, 62196, 70300, 79079, 88560, 98770, 109736, 121485, 134044, 147440, 161700, 176851, 192920, 209934, 227920, 246905
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
COMMENTS
|
In a matchstick-built Star of David, this sequence with a leading 0 is the number of triangles larger than size 1 in one orientation; there is an equal number of 'inverted' triangles.
For such a star, A045946 gives the number of matches; A135453 gives the number of size=1 triangles and A299965 gives the total number of triangles. For the triangle analogs see A045943; for the hexagon analogs see A045949. (End)
|
|
LINKS
|
|
|
FORMULA
|
a(n) = -a(-n) = binomial(3*n + 1,3) = 1/6*(3*n + 1)*(3*n)*(3*n - 1).
G.f.: (4*x + 19*x^2 + 4*x^3)/(1 - x)^4 = 4*x + 35*x^2 + 120*x^3 + ....
Sum_{n>=1} 1/a(n) = 3*log(3) - 3.
Sum_{n>=1} (-1)^n/a(n) = 4*log(2) - 3.
|
|
MAPLE
|
seq(binomial(3*n+1, 3), n = 1..38);
|
|
MATHEMATICA
|
LinearRecurrence[{4, -6, 4, -1}, {4, 35, 120, 286}, 40] (* Harvey P. Dale, Jan 11 2015 *)
|
|
PROG
|
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|