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%I M1039 N0390 #29 Oct 23 2015 19:38:40
%S 2,4,7,11,16,21,28,35,43,52,62,72,84,96,109,123,138,153,170,187,205,
%T 224,244,264,286,308,331,355,380,405,432,459,487,516,546,576,608,640,
%U 673,707,742,777,814,851,889,928,968,1008,1050,1092,1135,1179,1224,1269
%N Number of integer points in a certain quadrilateral scaled by a factor of n.
%C 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
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H E. Ehrhart, <a href="http://dx.doi.org/10.1515/crll.1967.226.1">Sur un problème de géométrie diophantienne linéaire I</a>, (Polyèdres et réseaux), J. Reine Angew. Math. 226 1967 1-29. MR0213320 (35 #4184).
%H E. Ehrhart, <a href="/A002789/a002789.pdf">Sur un problème de géométrie diophantienne linéaire I, (Polyèdres et réseaux)</a>, J. Reine Angew. Math. 226 1967 1-29. MR0213320 (35 #4184). [Annotated scanned copy of pages 16 and 22 only]
%H E. Ehrhart, <a href="/A002789/a002789_1.pdf">Sur un problème de géométrie diophantienne linéaire II. Systemes diophantiens lineaires</a>, J. Reine Angew. Math. 227 1967 25-49. [Annotated scanned copy of pages 47-49 only]
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Ehrhart_polynomial">Ehrhart polynomial</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,-1,-1,1).
%F 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
%F 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
%e 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 + ...
%t a[ n_] := Quotient[ 7 + 12 n + 5 n^2, 12]; (* _Michael Somos_, May 22 2014 *)
%t 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 *)
%o (PARI) {a(n) = (7 + 12*n + 5*n^2) \ 12}; /* _Michael Somos_, May 22 2014 */
%o (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 */
%Y Cf. A168668, A242771, A242774.
%K nonn
%O 1,1
%A _N. J. A. Sloane_
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