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
From Mircea Dan Rus, Jul 11 2020: (Start)
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.
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(End)
LINKS
Clark Kimberling, Table of n, a(n) for n = 1..200
Teofil Bogdan and Mircea Rus, Numărând dreptunghiuri pe foaia de matematică (in Romanian). Gazeta Matematică, seria B, 2020 (6-7-8), pp. 281-288.
Teofil Bogdan and Mircea Dan Rus, Counting the lattice rectangles inside Aztec diamonds and square biscuits, arXiv:2007.13472 [math.CO], 2020.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
FORMULA
MAPLE
A213840:=n->n*(1 + n)*(3 - 4*n + 4*n^2)/6: seq(A213840(n), n=1..50); # Wesley Ivan Hurt, Sep 16 2017
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
(Magma) [n*(1+n)*(3-4*n+4*n^2)/6: n in [1..60]]; // Vincenzo Librandi, Aug 01 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jul 05 2012
EXTENSIONS
Edited (with simpler definition) by N. J. A. Sloane, Sep 19 2017
STATUS
approved