|
|
A213840
|
|
a(n) = n*(1 + n)*(3 - 4*n + 4*n^2)/6.
|
|
3
|
|
|
1, 11, 54, 170, 415, 861, 1596, 2724, 4365, 6655, 9746, 13806, 19019, 25585, 33720, 43656, 55641, 69939, 86830, 106610, 129591, 156101, 186484, 221100, 260325, 304551, 354186, 409654, 471395, 539865, 615536, 698896, 790449, 890715, 1000230, 1119546, 1249231
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
Antidiagonal sums of the convolution array A213838.
The sequence is the binomial transform of (1, 10, 33, 40, 16, 0, 0, 0, ...). - Gary W. Adamson, Jul 31 2015
a(n) is also the number of rectangles in a square biscuit of order n, which is obtained by stacking 2n-1 rows with their centers vertically aligned which consist successively of 1, 3, ..., 2n-3, 2n-1, 2n-3, ..., 3, 1 consecutive unit lattice squares. The order 2 and 3 square biscuits are shown below which contain 11 and 54 rectangles respectively.
__
__ __|__|__
__|__|__ __|__|__|__|__
|__|__|__| |__|__|__|__|__|
|__| |__|__|__|
|__|
(End)
|
|
LINKS
|
|
|
FORMULA
|
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: x*(1 + 3*x)^2/(1 - x)^5.
|
|
MAPLE
|
|
|
MATHEMATICA
|
Table[n (1 + n) (3 - 4 n + 4 n^2)/6, {n, 50}] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {1, 11, 54, 170, 415}, 40] (* Vincenzo Librandi, Aug 01 2015 *)
|
|
PROG
|
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|