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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
OFFSET
1,3
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
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: A245484 A215503 A371835 * A119546 A173204 A229820
KEYWORD
nonn,tabl
AUTHOR
R. H. Hardin, Sep 13 2011
STATUS
approved

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Last modified September 20 12:37 EDT 2024. Contains 376072 sequences. (Running on oeis4.)