A213840 a(n) = n*(1 + n)*(3 - 4*n + 4*n^2)/6.

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

Offset: 1

Author

Clark Kimberling, Jul 05 2012

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A000332, A002417, A213838.

First differences of A271870. - J. M. Bergot, Aug 29 2016

Programs

Magma

[n*(1+n)*(3-4*n+4*n^2)/6: n in [1..60]]; // Vincenzo Librandi, Aug 01 2015

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 *)

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.

From Mircea Dan Rus, Aug 26 2020: (Start)

a(n) = A000332(n+3) + 6*A000332(n+2) + 9*A000332(n+1).

a(n) = A002417(n) + 3*A002417(n-1). (End)

Extensions

Edited (with simpler definition) by N. J. A. Sloane, Sep 19 2017