A242771 Number of integer points in a certain quadrilateral scaled by a factor of n (another version).

0, 0, 1, 3, 6, 9, 14, 19, 25, 32, 40, 48, 58, 68, 79, 91, 104, 117, 132, 147, 163, 180, 198, 216, 236, 256, 277, 299, 322, 345, 370, 395, 421, 448, 476, 504, 534, 564, 595, 627, 660, 693, 728, 763, 799, 836, 874, 912, 952, 992, 1033, 1075, 1118, 1161, 1206

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).

A014311 \ A333255 ranks these compositions.

A140106 is the unordered version.

A337484 is the case not strictly decreasing either.

A337698 counts these compositions of any length, with complement A000009.

A001399(n-6) counts unordered strict triples.

A001523 counts unimodal compositions, with complement A115981.

A007318 and A097805 count compositions by length.

A069905 counts unordered triples.

A218004 counts strictly increasing or weakly decreasing compositions.

A337483 counts triples either weakly increasing or weakly decreasing.

Cf. A332834, A337461, A337481, A337482, A337604.

(End)

Links

Table of n, a(n) for n=1..55.

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

G.f.: x^3 * (1 + 2*x + 2*x^2) / (1 - x - x^2 + x^4 + x^5 - x^6) = (x^3 + x^4 + x^5 + 2*x^7) / ((1 - x)^2 * (1 - x^6)).

a(n) = floor( A147874(n) / 12).

a(-n) = A002789(n).

a(n+1) - a(n) = A010761(n).

For n >= 6, a(n) = A000217(n-2) - A001399(n-6). - Gus Wiseman, Oct 18 2020

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

Cf. A002789, A010761, A147874.

Cf. A000212, A000217, A001840, A046691, A128422, A156040.

Sequence in context: A261229 A184015 A225282 * A310169 A310170 A310171

Adjacent sequences: A242768 A242769 A242770 * A242772 A242773 A242774

Keyword

nonn

Author

Michael Somos, May 22 2014

Status

approved