This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A242771 #31 Sep 08 2022 08:46:08
%S A242771 0,0,1,3,6,9,14,19,25,32,40,48,58,68,79,91,104,117,132,147,163,180,
%T A242771 198,216,236,256,277,299,322,345,370,395,421,448,476,504,534,564,595,
%U A242771 627,660,693,728,763,799,836,874,912,952,992,1033,1075,1118,1161,1206
%N A242771 Number of integer points in a certain quadrilateral scaled by a factor of n (another version).
%C A242771 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).
%C A242771 From _Gus Wiseman_, Oct 18 2020: (Start)
%C A242771 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:
%C A242771 (1,1,1) (1,1,2) (1,1,3) (1,1,4) (1,1,5)
%C A242771 (1,2,1) (1,2,2) (1,3,2) (1,3,3)
%C A242771 (2,1,1) (1,3,1) (1,4,1) (1,4,2)
%C A242771 (2,1,2) (2,1,3) (1,5,1)
%C A242771 (2,2,1) (2,2,2) (2,1,4)
%C A242771 (3,1,1) (2,3,1) (2,2,3)
%C A242771 (3,1,2) (2,3,2)
%C A242771 (3,2,1) (2,4,1)
%C A242771 (4,1,1) (3,1,3)
%C A242771 (3,2,2)
%C A242771 (3,3,1)
%C A242771 (4,1,2)
%C A242771 (4,2,1)
%C A242771 (5,1,1)
%C A242771 A001399(n-6) counts the complement (unordered strict triples).
%C A242771 A014311 \ A333255 ranks these compositions.
%C A242771 A140106 is the unordered version.
%C A242771 A337484 is the case not strictly decreasing either.
%C A242771 A337698 counts these compositions of any length, with complement A000009.
%C A242771 A001399(n-6) counts unordered strict triples.
%C A242771 A001523 counts unimodal compositions, with complement A115981.
%C A242771 A007318 and A097805 count compositions by length.
%C A242771 A069905 counts unordered triples.
%C A242771 A218004 counts strictly increasing or weakly decreasing compositions.
%C A242771 A337483 counts triples either weakly increasing or weakly decreasing.
%C A242771 Cf. A332834, A337461, A337481, A337482, A337604.
%C A242771 (End)
%H A242771 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 A242771 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 A242771 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 A242771 Wikipedia, <a href="http://en.wikipedia.org/wiki/Ehrhart_polynomial">Ehrhart polynomial</a>
%H A242771 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,-1,-1,1).
%F A242771 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)).
%F A242771 a(n) = floor( A147874(n) / 12).
%F A242771 a(-n) = A002789(n).
%F A242771 a(n+1) - a(n) = A010761(n).
%F A242771 For n >= 6, a(n) = A000217(n-2) - A001399(n-6). - _Gus Wiseman_, Oct 18 2020
%e A242771 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 + ...
%t A242771 a[ n_] := Quotient[ 7 - 12 n + 5 n^2, 12];
%t A242771 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]];
%t A242771 LinearRecurrence[{1,1,0,-1,-1,1},{0,0,1,3,6,9},90] (* _Harvey P. Dale_, May 28 2015 *)
%t A242771 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],!Less@@#&]],{n,0,15}] (* _Gus Wiseman_, Oct 18 2020 *)
%o A242771 (PARI) {a(n) = (7 - 12*n + 5*n^2) \ 12};
%o A242771 (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))};
%o A242771 (Magma) [Floor((5*n-7)*(n-1)/12): n in [1..60]]; // _Vincenzo Librandi_, Jun 27 2015
%Y A242771 Cf. A002789, A010761, A147874.
%Y A242771 Cf. A000212, A000217, A001840, A046691, A128422, A156040.
%K A242771 nonn
%O A242771 1,4
%A A242771 _Michael Somos_, May 22 2014