login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A195220 T(n,k) is the number of lower triangles of an n X n integer array with each element differing from all of its diagonal, vertical, antidiagonal and horizontal neighbors by k or less and triangles differing by a constant counted only once. 13
1, 1, 7, 1, 19, 91, 1, 37, 1047, 2277, 1, 61, 5453, 176471, 111031, 1, 91, 18903, 3395245, 92031109, 10654607, 1, 127, 51205, 31640829, 9032683465, 149824887097, 2021888119, 1, 169, 117585, 189677411, 289301569283, 103565705397639 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Table starts

        1            1               1                 1                  1

        7           19              37                61                 91

       91         1047            5453             18903              51205

     2277       176471         3395245          31640829          189677411

   111031     92031109      9032683465      289301569283      4677360495205

10654607 149824887097 103565705397639 14572563308953245 774355028021195459

T(n,k) is the number of integer lattice points in kP where P is a (n*(n+1)/2-1)-dimensional polytope with vertices whose coordinates are all in {-1,0,1}.  Therefore it is an Ehrhart polynomial in k, with degree n*(n+1)/2-1 and rational coefficients. - Robert Israel, Oct 06 2019

LINKS

R. H. Hardin, Table of n, a(n) for n = 1..86

Wikipedia, Ehrhart polynomial

FORMULA

Empirical for rows:

T(1,k) = 1

T(2,k) = 3*k^2 + 3*k + 1

T(3,k) = (301/30)*k^5 + (301/12)*k^4 + (88/3)*k^3 + (227/12)*k^2 + (199/30)*k + 1

T(4,k) = (1207573/30240)*k^9 + (1207573/6720)*k^8 + (1000157/2520)*k^7 + (264247/480)*k^6 + (754417/1440)*k^5 + (338651/960)*k^4 + (2533393/15120)*k^3 + (90763/1680)*k^2 + (901/84)*k + 1

T(5,k) = (3508493543/18345600)*k^14 + (3508493543/2620800)*k^13 + (1116775769537/239500800)*k^12 + (422094048023/39916800)*k^11 + (377328209183/21772800)*k^10 + (78475421219/3628800)*k^9 + (1073748492569/50803200)*k^8 + (19848770813/1209600)*k^7 + (221251862417/21772800)*k^6 + (18121075223/3628800)*k^5 + (10435002133/5443200)*k^4 + (505904317/907200)*k^3 + (8793472607/75675600)*k^2 + (1397863/90090)*k + 1

EXAMPLE

Some solutions for n=6, k=5:

   0                  0                  0                  0

   4  4               2  2               2  1               4  5

   6  7  7            7  6  5           -3 -2  1            5  8  7

  10  8 12  7         4  7  6  1        -6 -1 -4 -2         8  9  5  7

  10 12 11 12  9      2  3  5  1  0     -1 -1 -1 -1 -2      5  5  8  9  8

   7  7  8 12  9  5   1  3  5  2  4  5  -6 -3  0  0  1 -3   0  3  8  8 10  6

CROSSREFS

Row 2 is A003215.

Sequence in context: A028325 A245484 A215503 * A119546 A173204 A229820

Adjacent sequences:  A195217 A195218 A195219 * A195221 A195222 A195223

KEYWORD

nonn,tabl

AUTHOR

R. H. Hardin, Sep 13 2011

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 7 23:48 EDT 2021. Contains 343652 sequences. (Running on oeis4.)