A002789 -id:A002789 - cp's OEIS frontend

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

Michael Somos, May 22 2014

nonn

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)

			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 + ...

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).

Cf. A002789, A010761, A147874.

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

[Floor((5*n-7)*(n-1)/12): n in [1..60]]; // Vincenzo Librandi, Jun 27 2015

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 *)

PARI
```
{a(n) = (7 - 12*n + 5*n^2) \ 12};
```

{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))};

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

A002798 a(n) = a(n-1) + a(n-2) - a(n-3).

Original entry on oeis.org

18, 45, 69, 96, 120, 147, 171, 198, 222, 249, 273, 300, 324, 351, 375, 402, 426, 453, 477, 504, 528, 555, 579, 606, 630, 657, 681, 708, 732, 759, 783, 810, 834, 861, 885, 912, 936, 963, 987, 1014, 1038, 1065, 1089

Offset: 1

N. J. A. Sloane and Simon Plouffe

nonn

The old definition was a(n) = a(n-2)+a(n-3)-a(n-5).

The following applies to this sequence and also to all sequences of the form a(n) = a(n-1) + a(n-2) - a(n-3), regardless of initial values: (a(n+3i) + a(n))/(a(n+2i) + a(n+i)) = 1, as long as a(n+2i) + a(n+i) != 0. - Klaus Purath, Jun 05 2024

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).

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]
E. Ehrhart, Sur un problème de géométrie diophantienne linéaire II, (Systèmes diophantiens linéaires), J. Reine Angew. Math. 227 1967 25-49.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A007310, A047208.

A002798:=3*(6+9*z+2*z**2)/(z+1)/(z-1)**2; # Simon Plouffe in his 1992 dissertation

LinearRecurrence[{1,1,-1},{18,45,69},50] (* Harvey P. Dale, Sep 17 2023 *)

a(n) = 6*A007310(n) + 3*A047208(n).

a(n) = (51*n - 12)/2 - 3*(1 - (-1)^n)/4 = 2*a(n-1) - a(n-2) + 3(-1)^n. - Klaus Purath, Jun 05 2024

Definition simplified by Ray Chandler. - N. J. A. Sloane, Mar 07 2024

A242774 a(n) = ceiling( n / 2 ) + ceiling( n / 3 ).

Original entry on oeis.org

2, 2, 3, 4, 5, 5, 7, 7, 8, 9, 10, 10, 12, 12, 13, 14, 15, 15, 17, 17, 18, 19, 20, 20, 22, 22, 23, 24, 25, 25, 27, 27, 28, 29, 30, 30, 32, 32, 33, 34, 35, 35, 37, 37, 38, 39, 40, 40, 42, 42, 43, 44, 45, 45, 47, 47, 48, 49, 50, 50, 52, 52, 53, 54, 55, 55, 57

Offset: 1

Michael Somos, May 22 2014

			G.f. = 2*x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 5*x^6 + 7*x^7 + 7*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (0,1,1,0,-1).
Index entries for two-way infinite sequences

Cf. A002789, A010761, A110654.

Cf. A000035, A010872.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(2+2*x+x^2)/(1-x^2-x^3+x^5)));  // G. C. Greubel, Aug 06 2018

A242774:=n->ceil(n/2)+ceil(n/3): seq(A242774(n), n=1..100); # Wesley Ivan Hurt, Apr 13 2016

a[ n_] := Ceiling[ n / 2 ] + Ceiling[ n / 3 ];
LinearRecurrence[{0, 1, 1, 0, -1}, {2, 2, 3, 4, 5}, 100] (* Vincenzo Librandi, Apr 15 2016 *)
Rest[CoefficientList[Series[x*(2+2*x+x^2)/(1-x^2-x^3+x^5), {x, 0, 50}], x]] (* G. C. Greubel, Aug 06 2018 *)

{a(n) = ceil( n / 2 ) + ceil( n / 3 )};

{a(n) = if( n<0, polcoeff( -(x^2 + 2*x^3 + 2*x^4) / ((1 - x^2) * (1 - x^3)) + x * O(x^-n), -n), polcoeff( (2*x + 2*x^2 + x^3) / ((1 - x^2) * (1 - x^3)) + x * O(x^n), n))};

G.f.: x * (2 + 2*x + x^2) / (1 - x^2 - x^3 + x^5) = (2*x + 2*x^2 + x^3) / ((1 - x^2) * (1 - x^3)).
a(n) = - A010761(-n) = 2 - a(1-n). a(n) = A002789(n) - A002789(n-1) for all n in Z.
a(n) = Sum_{k=1..n} A000035(k) + A000035(A010872(k)). - Benedict W. J. Irwin, Apr 13 2016
E.g.f.: 5*x*exp(x)/6 - exp(-x)/4 + 7*exp(x)/12 + sin(sqrt(3)*x/2)*exp(-x/2)/(3*sqrt(3)) - cos(sqrt(3)*x/2)*exp(-x/2)/3. - Ilya Gutkovskiy, Apr 13 2016

A002797 Number of solutions to a linear inequality.

Original entry on oeis.org

3, 2, 5, 9, 17, 27, 40, 55, 73, 94, 117, 143, 171, 203, 236, 273, 311, 354, 397, 445, 493, 547, 600, 659, 717, 782, 845, 915, 983, 1059, 1132, 1213, 1291, 1378, 1461, 1553, 1641, 1739, 1832, 1935, 2033, 2142, 2245, 2359, 2467, 2587, 2700

Offset: 0

N. J. A. Sloane

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).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
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]
Ehrhart, E.; Sur un problème de géométrie diophantienne linéaire II. Systemes diophantiens lineaires, J. Reine Angew. Math. 227 1967 25-49.
Index entries for linear recurrences with constant coefficients, signature (1, 1, -1, 1, -1, -1, 1).

Vec(-(5*x^6+7*x^5+2*x^4+5*x^3-x+3)/((x^2+1)*(x+1)^2*(x-1)^3) + O(x^50)) \\ Michel Marcus, Jan 26 2015

a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) - a(n-5) - a(n-6) + a(n-7). - Sean A. Irvine, Aug 20 2014

G.f.: -(5*x^6+7*x^5+2*x^4+5*x^3-x+3)/((x^2+1)*(x+1)^2*(x-1)^3). - Alois P. Heinz, Aug 20 2014

Initial term, missing a(9), and more terms from Sean A. Irvine, Aug 20 2014

cp's OEIS Frontend

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

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A002798 a(n) = a(n-1) + a(n-2) - a(n-3).

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A242774 a(n) = ceiling( n / 2 ) + ceiling( n / 3 ).

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Magma

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A002797 Number of solutions to a linear inequality.

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PARI

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Extensions