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T(n,k) is the number of -k..k arrays of n elements with first, second and third differences also in -k..k.
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%I #14 Jun 30 2019 20:25:03

%S 3,5,7,7,19,13,9,37,57,19,11,61,153,127,27,13,91,323,475,293,35,15,

%T 127,587,1279,1509,663,47,17,169,967,2833,5205,4763,1517,65,19,217,

%U 1483,5509,14063,21093,15101,3459,91,21,271,2157,9739,32267,69573,85771,47889,7905

%N T(n,k) is the number of -k..k arrays of n elements with first, second and third differences also in -k..k.

%C Table starts

%C ...3.....5......7.......9.......11........13........15.........17.........19

%C ...7....19.....37......61.......91.......127.......169........217........271

%C ..13....57....153.....323......587.......967......1483.......2157.......3009

%C ..19...127....475....1279.....2833......5509......9739......16039......25003

%C ..27...293...1509....5205....14063.....32267.....65773.....122709.....213697

%C ..35...663...4763...21093....69573....188505....443169.....936715....1822729

%C ..47..1517..15101...85771...345241...1104357...2993875....7169025...15586785

%C ..65..3459..47889..348841..1713419...6471075..20229855...54878469..133314467

%C ..91..7905.151833.1418711..8503671..37917347.136692527..420086101.1140231725

%C .129.18051.481519.5769945.42203951.222179581.923636217.3215726871.9752442535

%C For fixed n, T(n,k) is the number of lattice points in k*C(n) where C(n) is a certain polytope in R^n whose vertices have rational coefficients. Therefore row n of the table is an Ehrhart quasi-polynomial of degree <= n. - _Robert Israel_, Jun 28 2019

%H R. H. Hardin, <a href="/A202124/b202124.txt">Table of n, a(n) for n = 1..7269</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Ehrhart_polynomial#Ehrhart_quasi-polynomials">Ehrhart quasi-polynomials</a>

%e Some solutions for n=6, k=5:

%e 3 2 -1 -5 -3 5 1 4 -2 0 3 4 -3 -3 -3 -5

%e -2 4 -2 0 -1 5 -3 -1 -2 1 -2 1 2 -4 -4 -4

%e -5 3 -2 4 1 4 -2 -2 1 3 -5 -1 3 -3 -3 0

%e -5 -1 -1 4 2 5 2 0 2 3 -4 -3 2 0 1 2

%e -2 -3 -3 4 4 5 5 2 1 1 -3 -1 1 3 5 3

%e 3 -4 -5 0 5 0 5 5 1 0 0 1 2 2 5 2

%Y Row 2 is A003215.

%Y Row 3 is A007202.

%K nonn,tabl

%O 1,1

%A _R. H. Hardin_, Dec 11 2011