OFFSET
1,4
COMMENTS
The quadrilateral is given by four vertices [(1/2, 1/3), (0, 1), (0, 0), (1, 0)] as an example on page 22 of Ehrhart 1967. Here the open line segment from (1/2, 1/3) to (0, 1) is included but the rest of the boundary is not. The sequence is denoted by d'(n).
From Gus Wiseman, Oct 18 2020: (Start)
Also the number of ordered triples of positive integers summing to n that are not strictly increasing. For example, the a(3) = 1 through a(7) = 14 triples are:
(1,1,1) (1,1,2) (1,1,3) (1,1,4) (1,1,5)
(1,2,1) (1,2,2) (1,3,2) (1,3,3)
(2,1,1) (1,3,1) (1,4,1) (1,4,2)
(2,1,2) (2,1,3) (1,5,1)
(2,2,1) (2,2,2) (2,1,4)
(3,1,1) (2,3,1) (2,2,3)
(3,1,2) (2,3,2)
(3,2,1) (2,4,1)
(4,1,1) (3,1,3)
(3,2,2)
(3,3,1)
(4,1,2)
(4,2,1)
(5,1,1)
A001399(n-6) counts the complement (unordered strict triples).
A140106 is the unordered version.
A337484 is the case not strictly decreasing either.
A001399(n-6) counts unordered strict triples.
A069905 counts unordered triples.
A218004 counts strictly increasing or weakly decreasing compositions.
A337483 counts triples either weakly increasing or weakly decreasing.
(End)
LINKS
E. Ehrhart, Sur un problème de géométrie diophantienne linéaire I, (Polyèdres et réseaux), J. Reine Angew. Math. 226 1967 1-29. MR0213320 (35 #4184).
E. Ehrhart, Sur un problème de géométrie diophantienne linéaire I, (Polyèdres et réseaux), J. Reine Angew. Math. 226 1967 1-29. MR0213320 (35 #4184). [Annotated scanned copy of pages 16 and 22 only]
E. Ehrhart, Sur un problème de géométrie diophantienne linéaire II. Systemes diophantiens lineaires, J. Reine Angew. Math. 227 1967 25-49. [Annotated scanned copy of pages 47-49 only]
Wikipedia, Ehrhart polynomial
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).
FORMULA
EXAMPLE
G.f. = x^3 + 3*x^4 + 6*x^5 + 9*x^6 + 14*x^7 + 19*x^8 + 25*x^9 + 32*x^10 + ...
MATHEMATICA
a[ n_] := Quotient[ 7 - 12 n + 5 n^2, 12];
a[ n_] := With[ {o = Boole[ 0 < n], c = Boole[ 0 >= n], m = Abs@n}, Length @ FindInstance[ 0 < c + x && 0 < c + y && (2 x < c + m && 4 x + 3 y < o + 3 m || m < o + 2 x && 2 x + 3 y < c + 2 m), {x, y}, Integers, 10^9]];
LinearRecurrence[{1, 1, 0, -1, -1, 1}, {0, 0, 1, 3, 6, 9}, 90] (* Harvey P. Dale, May 28 2015 *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n, {3}], !Less@@#&]], {n, 0, 15}] (* Gus Wiseman, Oct 18 2020 *)
PROG
(PARI) {a(n) = (7 - 12*n + 5*n^2) \ 12};
(PARI) {a(n) = if( n<0, polcoeff( x * (2 + x^2 + x^3 + x^4) / ((1 - x)^2 * (1 - x^6)) + x * O(x^-n), -n), polcoeff( x^3 * (1 + x + x^2 + 2*x^4) / ((1 - x)^2 * (1 - x^6)) + x * O(x^n), n))};
(Magma) [Floor((5*n-7)*(n-1)/12): n in [1..60]]; // Vincenzo Librandi, Jun 27 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, May 22 2014
STATUS
approved