A002789 Number of integer points in a certain quadrilateral scaled by a factor of n. (Formerly M1039 N0390)

2, 4, 7, 11, 16, 21, 28, 35, 43, 52, 62, 72, 84, 96, 109, 123, 138, 153, 170, 187, 205, 224, 244, 264, 286, 308, 331, 355, 380, 405, 432, 459, 487, 516, 546, 576, 608, 640, 673, 707, 742, 777, 814, 851, 889, 928, 968, 1008, 1050, 1092, 1135, 1179, 1224, 1269

Offset

1,1

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 closed line segment from (1/2, 1/3) to (0, 1) is not included but the rest of the boundary is. The sequence is denoted by d(n). - Michael Somos, May 22 2014

References

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Table of n, a(n) for n=1..54.

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 * (2 + 2*x + x^2) / (1 - x - x^2 + x^4 + x^5 - x^6) = (2*x + x^3 + x^4 + x^5) / ((1 - x)^2 * (1 - x^6)). - Michael Somos, May 22 2014

a(n) = floor( A168668(n+1) / 12), a(n) = A242771(-n), a(n) - a(n-1) = A242774(n) for all n in Z. - Michael Somos, May 22 2014

Example

G.f. = 2*x + 4*x^2 + 7*x^3 + 11*x^4 + 16*x^5 + 21*x^6 + 28*x^7 + 35*x^8 + ...

Mathematica

a[ n_] := Quotient[ 7 + 12 n + 5 n^2, 12]; (* Michael Somos, May 22 2014 *)
a[ n_] := Length @ With[{o = Boole[ 0 < n], c = Boole[ 0 >= n], m = Abs@n}, FindInstance[ 0 < o + x && 0 < o + y && (2 x < o + m && 4 x + 3 y < c + 3 m || m < c + 2 x && 2 x + 3 y < o + 2 m), {x, y}, Integers, 10^9]]; (* Michael Somos, May 22 2014 *)

Prog

(PARI) {a(n) = (7 + 12*n + 5*n^2) \ 12}; /* Michael Somos, May 22 2014 */
(PARI) {a(n) = if( n<0, polcoeff( x^3 * (1 + x + x^2 + 2*x^4) / ((1 - x)^2 * (1 - x^6)) + x * O(x^-n), -n), polcoeff( x * (2 + x^2 + x^3 + x^4) / ((1 - x)^2 * (1 - x^6)) + x * O(x^n), n))}; /* Michael Somos, May 22 2014 */

Crossrefs

Cf. A168668, A242771, A242774.

Sequence in context: A022783 A025708 A024673 * A083204 A061784 A005311

Adjacent sequences: A002786 A002787 A002788 * A002790 A002791 A002792

Keyword

nonn

Author

N. J. A. Sloane

Status

approved