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

3, 5, 7, 7, 19, 13, 9, 37, 57, 19, 11, 61, 153, 127, 27, 13, 91, 323, 475, 293, 35, 15, 127, 587, 1279, 1509, 663, 47, 17, 169, 967, 2833, 5205, 4763, 1517, 65, 19, 217, 1483, 5509, 14063, 21093, 15101, 3459, 91, 21, 271, 2157, 9739, 32267, 69573, 85771, 47889, 7905

Offset: 1

Author

R. H. Hardin, Dec 11 2011

Keywords

• nonn
• tabl

Comments

Table starts

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

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

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

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

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

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

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

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

..91..7905.151833.1418711..8503671..37917347.136692527..420086101.1140231725

.129.18051.481519.5769945.42203951.222179581.923636217.3215726871.9752442535

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

Examples

			Some solutions for n=6, k=5:
   3  2 -1 -5 -3  5  1  4 -2  0  3  4 -3 -3 -3 -5
  -2  4 -2  0 -1  5 -3 -1 -2  1 -2  1  2 -4 -4 -4
  -5  3 -2  4  1  4 -2 -2  1  3 -5 -1  3 -3 -3  0
  -5 -1 -1  4  2  5  2  0  2  3 -4 -3  2  0  1  2
  -2 -3 -3  4  4  5  5  2  1  1 -3 -1  1  3  5  3
   3 -4 -5  0  5  0  5  5  1  0  0  1  2  2  5  2
		

Links

• R. H. Hardin, Table of n, a(n) for n = 1..7269
• Wikipedia, Ehrhart quasi-polynomials

Crossrefs

Row 2 is A003215.

Row 3 is A007202.