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A202124 T(n,k) is the number of -k..k arrays of n elements with first, second and third differences also in -k..k.

%I A202124 #14 Jun 30 2019 20:25:03
%S A202124 3,5,7,7,19,13,9,37,57,19,11,61,153,127,27,13,91,323,475,293,35,15,
%T A202124 127,587,1279,1509,663,47,17,169,967,2833,5205,4763,1517,65,19,217,
%U A202124 1483,5509,14063,21093,15101,3459,91,21,271,2157,9739,32267,69573,85771,47889,7905
%N A202124 T(n,k) is the number of -k..k arrays of n elements with first, second and third differences also in -k..k.
%C A202124 Table starts
%C A202124 ...3.....5......7.......9.......11........13........15.........17.........19
%C A202124 ...7....19.....37......61.......91.......127.......169........217........271
%C A202124 ..13....57....153.....323......587.......967......1483.......2157.......3009
%C A202124 ..19...127....475....1279.....2833......5509......9739......16039......25003
%C A202124 ..27...293...1509....5205....14063.....32267.....65773.....122709.....213697
%C A202124 ..35...663...4763...21093....69573....188505....443169.....936715....1822729
%C A202124 ..47..1517..15101...85771...345241...1104357...2993875....7169025...15586785
%C A202124 ..65..3459..47889..348841..1713419...6471075..20229855...54878469..133314467
%C A202124 ..91..7905.151833.1418711..8503671..37917347.136692527..420086101.1140231725
%C A202124 .129.18051.481519.5769945.42203951.222179581.923636217.3215726871.9752442535
%C A202124 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 A202124 R. H. Hardin, <a href="/A202124/b202124.txt">Table of n, a(n) for n = 1..7269</a>
%H A202124 Wikipedia, <a href="https://en.wikipedia.org/wiki/Ehrhart_polynomial#Ehrhart_quasi-polynomials">Ehrhart quasi-polynomials</a>
%e A202124 Some solutions for n=6, k=5:
%e A202124    3  2 -1 -5 -3  5  1  4 -2  0  3  4 -3 -3 -3 -5
%e A202124   -2  4 -2  0 -1  5 -3 -1 -2  1 -2  1  2 -4 -4 -4
%e A202124   -5  3 -2  4  1  4 -2 -2  1  3 -5 -1  3 -3 -3  0
%e A202124   -5 -1 -1  4  2  5  2  0  2  3 -4 -3  2  0  1  2
%e A202124   -2 -3 -3  4  4  5  5  2  1  1 -3 -1  1  3  5  3
%e A202124    3 -4 -5  0  5  0  5  5  1  0  0  1  2  2  5  2
%Y A202124 Row 2 is A003215.
%Y A202124 Row 3 is A007202.
%K A202124 nonn,tabl
%O A202124 1,1
%A A202124 _R. H. Hardin_, Dec 11 2011