OFFSET
1,1
LINKS
Christopher Hunt Gribble, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,2,-3,3,-1).
FORMULA
a(n) = sum_{j=0..n-1-floor(n/3)} ((4*n-6*j+1-(-1)^j)/4)*((4*n-6*j+3+(-1)^j)/4).
a(n) = (4*n^3+15*n^2+17*n-6*floor(n/3))/18.
G.f.: x*(x^2+x+2) / ((x-1)^4*(x^2+x+1)). - Colin Barker, Apr 26 2014
EXAMPLE
The bi-symmetric triangle of 1 X 1 squares of height 5 is:
___
_|_|_|_
_|_|_|_|_|_
_|_|_|_|_|_|_|_
_|_|_|_|_|_|_|_|_|_
|_|_|_|_|_|_|_|_|_|_|
.
No. of positions in which a 1 X 1 square can be placed = 2 + 4 + 6 + 8 + 10 = 30.
No. of positions in which a 2 X 2 square can be placed = 1 + 3 + 5 + 7 = 16.
No. of positions in which a 3 X 3 square can be placed = 2 + 4 = 6.
No. of positions in which a 4 X 4 square can be placed = 1.
Thus, a(5) = 30 + 16 + 6 + 1 = 53.
MAPLE
a := proc (n::integer)::integer;
(2/9)*n^3+(5/6)*n^2+(17/18)*n-(1/3)*floor((1/3)*n)
end proc:
seq(a(n), n = 1..60);
PROG
(PARI) Vec(x*(x^2+x+2)/((x-1)^4*(x^2+x+1)) + O(x^100)) \\ Colin Barker, Apr 26 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Christopher Hunt Gribble and Luce ETIENNE, Apr 24 2014
STATUS
approved