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A242771 Number of integer points in a certain quadrilateral scaled by a factor of n (another version). 1
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 (list; graph; refs; listen; history; text; internal format)
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).

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).

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 *)

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.

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

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Last modified August 8 08:20 EDT 2020. Contains 336292 sequences. (Running on oeis4.)