A046691 -id:A046691 - 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

A248158 Expansion of (1 - 2*x^2)/(1 + x)^3. Second column of Riordan triangle A248156.

Original entry on oeis.org

1, -3, 4, -4, 3, -1, -2, 6, -11, 17, -24, 32, -41, 51, -62, 74, -87, 101, -116, 132, -149, 167, -186, 206, -227, 249, -272, 296, -321, 347, -374, 402, -431, 461, -492, 524, -557, 591, -626, 662, -699, 737, -776, 816, -857, 899, -942, 986

Offset: 0

Wolfdieter Lang, Oct 05 2014

This is the column k=1 sequence of the Riordan triangle A248156 without a leading zero.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-3,-3,-1).

Cf. A046691, A148157, A248156(n+1,1).

[(-1)^n*(2+5*n-n^2)/2: n in [0..60]]; // G. C. Greubel, May 30 2025

Table[(-1)^n*(2+5*n-n^2)/2, {n,0,60}] (* G. C. Greubel, May 30 2025 *)

def A248158(n): return (-1)**n*(2+5*n-n**2)//2
print([A248158(n) for n in range(51)]) # G. C. Greubel, May 30 2025

O.g.f.: (1 - 2*x^2)/(1 + x)^3 = -2/(1 + x) + 4/(1 + x)^2 - 1/(1 + x)^3.
a(n) = (-1)^n*(4*(2*n+1) - (n+1)*(n+2))/2, n >= 0.
a(n) = -3*(a(n-1) + a(n-2)) - a(n-3), n >= 3 with a(0) = 1, a(1) = -3 and a(2) = 4.
From R. J. Mathar, Mar 13 2021: (Start)
a(n) = (-1)^(n+1)*A046691(n-5).
a(n) + a(n+1) = A248157(n+1). (End)
E.g.f.: (1/2)*(2 - 4*x - x^2)*exp(-x). - G. C. Greubel, May 30 2025

A060577 Number of homeomorphically irreducible general graphs on 2 labeled nodes and with n edges.

Original entry on oeis.org

1, 1, 4, 6, 11, 17, 24, 32, 41, 51, 62, 74, 87, 101, 116, 132, 149, 167, 186, 206, 227, 249, 272, 296, 321, 347, 374, 402, 431, 461, 492, 524, 557, 591, 626, 662, 699, 737, 776, 816, 857, 899, 942, 986, 1031, 1077, 1124, 1172, 1221, 1271, 1322, 1374, 1427

Offset: 0

Vladeta Jovovic, Apr 04 2001

A homeomorphically irreducible general graph is a graph with multiple edges and loops and without nodes of degree 2.

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983.

V. Jovovic, Generating functions for homeomorphically irreducible general graphs on n labeled nodes
V. Jovovic, Recurrences for the numbers of homeomorphically irreducible general graphs on m labeled nodes and n edges
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A003514, A046691, A060516, A060533-A060537, A060576-A060581.

gf := (2*x^5 - 4*x^4 + 4*x^3 - 4*x^2 + 2*x - 1)/(x - 1)^3: s := series(gf, x, 100): for i from 0 to 100 do printf(`%d,`,coeff(s,x,i)) od:

Join[{1, 1, 4}, Table[n (n + 3)/2 - 3, {n, 3, 60}]] (* Bruno Berselli, Aug 20 2015 *)

G.f.: (2*x^5 - 4*x^4 + 4*x^3 - 4*x^2 + 2*x - 1)/(x - 1)^3.
E.g.f. for homeomorphically irreducible general graphs with n nodes and k edges is (1 + x*y)^( - 1/2)*exp( - x*y/2 + x^2*y^2/4)*Sum_{k >= 0} 1/(1 - x)^binomial(k + 1, 2)*exp( - x^2*y*k^2/(2*(1 + x*y)) - x^2*y*k/2)*y^k/k!.
From Marco Ripà, Aug 20 2015: (Start)
a(n) = ceiling( (1/2)*(3*n^2 - 10*n + 9)/(n - 2) ) + floor( (3/2)*(n-1)^2 ) - n^2 + 3*n - 3 with n > 2, a(0) = a(1) = 1, a(2) = 4.
a(n) = n*(n + 3)/2 - 3 for n > 2.
a(n) = A046691(n-1) for n > 2. (End)

More terms from James Sellers, Apr 04 2001

A346376 a(n) = n^4 + 14n^3 + 63n^2 + 98*n + 28.

Original entry on oeis.org

28, 204, 604, 1348, 2580, 4468, 7204, 11004, 16108, 22780, 31308, 42004, 55204, 71268, 90580, 113548, 140604, 172204, 208828, 250980, 299188, 354004, 416004, 485788, 563980, 651228, 748204, 855604, 974148, 1104580, 1247668, 1404204, 1575004, 1760908, 1962780

Offset: 0

Lamine Ngom, Jul 14 2021

The product of eight consecutive positive integers can always be expressed as the difference of two squares: x^2 - y^2.
This sequence gives the x-values for each product. The y-values are A017113(n+4).
a(n) is always divisible by 4. In addition, we have (a(n)+16)/4 belongs to A028387.
Are 4 and 8 the unique values of k such that the product of k consecutive integers is always distant to upper square by a square?

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A239035, A017113, A028387, A046691.

a(n) = A239035(n)^2 - A017113(n+4)^2.
a(n) = 4*(A028387(A046691(n+2)) - 4).
G.f.: 4*(7 + 16*x - 34*x^2 + 22*x^3 - 5*x^4)/(1 - x)^5. - Stefano Spezia, Jul 14 2021

A179571 Number of permutations of 1..n+4 with the number moved left exceeding the number moved right by n.

Original entry on oeis.org

31, 66, 134, 267, 529, 1048, 2080, 4137, 8243, 16446, 32842, 65623, 131173, 262260, 524420, 1048725, 2097319, 4194490, 8388814, 16777443, 33554681, 67109136, 134218024, 268435777, 536871259, 1073742198, 2147484050, 4294967727

Offset: 1

R. H. Hardin, g.f. from R. J. Mathar in the Sequence Fans Mailing List, Jul 19 2010

nonn

Recurrence would also extend to an a(0) if the definition were made to exclude the identity permutation.

R. H. Hardin, Table of n, a(n) for n = 1..99

Cf. A083706.

Drop[Accumulate[Table[BitSet[n, (n + 2)], {n, 0, 100}]], 2] (* Vladimir Joseph Stephan Orlovsky, Jul 19 2011 *)

Empirical: a(n)=5*a(n-1)-9*a(n-2)+7*a(n-3)-2*a(n-4) ; G.f.: -x*(-31+89*x-83*x^2+26*x^3) / ( (2*x-1)*(x-1)^3 ).

Empirical: a(n) = (n^2+3*n-6)/2 +2^(n+4) = 2^(n+4)+A046691(n-1). - R. J. Mathar, May 26 2016

A356754 Triangle read by rows: T(n,k) = ((n-1)(n+2))/2 + 2k.

Original entry on oeis.org

2, 4, 6, 7, 9, 11, 11, 13, 15, 17, 16, 18, 20, 22, 24, 22, 24, 26, 28, 30, 32, 29, 31, 33, 35, 37, 39, 41, 37, 39, 41, 43, 45, 47, 49, 51, 46, 48, 50, 52, 54, 56, 58, 60, 62, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87

Offset: 1

Torlach Rush, Aug 25 2022

The first column of the triangle is the Lazy Caterer's sequence A000124.
Each subsequent column starts with A000124(n) + (2 * (n-1)).
The first downward diagonal is A046691(n).
Columns and downward diagonals of the triangle identify many sequences (possibly shifted) in the database. Examples can be found in crossrefs below.
The sum of the n-th upward diagonal of the triangle is A356288(n).

			Triangle T(n,k) begins:
  n\k   1   2   3   4   5   6   7   8   9  10  11  ...
   1:   2
   2:   4   6
   3:   7   9  11
   4:  11  13  15  17
   5:  16  18  20  22  24
   6:  22  24  26  28  30  32
   7:  29  31  33  35  37  39  41
   8:  37  39  41  43  45  47  49  51
   9:  46  48  50  52  54  56  58  60  62
  10:  56  58  60  62  64  66  68  70  72  74
  11:  67  69  71  73  75  77  79  81  83  85  87
  ...

Cf. A000124, A004120, A046691, A051938, A055999, A056000, A155212, A167487, A167499, A167614, A246172, A334563, A356288.

Table[((n-1)(n+2))/2+2k,{n,20},{k,n}]//Flatten (* Harvey P. Dale, May 26 2023 *)

def T(n, k): return ((n-1) * (n+2))//2 + 2*k
for n in range(1, 12):
    for k in range(1,(n+1)): print(T(n,k), end = ', ')

# Indexed as a linear sequence.
def a000124(n): return n*(n+1)//2 + 1
def a(n):
    l = m = 0
    for k in range(1,n):
        lc = a000124(k - 1)
        if n >= lc:
            l = lc
            m = k
        else: break
    return n + m + (n - l)

T(n,k) = ((n-1) * (n+2))/2 + 2*k.

T(n,k+1) = T(n,k) + 2, k < n.

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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A248158 Expansion of (1 - 2*x^2)/(1 + x)^3. Second column of Riordan triangle A248156.

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A060577 Number of homeomorphically irreducible general graphs on 2 labeled nodes and with n edges.

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Maple

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A346376 a(n) = n^4 + 14*n^3 + 63*n^2 + 98*n + 28.

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A179571 Number of permutations of 1..n+4 with the number moved left exceeding the number moved right by n.

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Programs

Mathematica

Formula

A356754 Triangle read by rows: T(n,k) = ((n-1)*(n+2))/2 + 2*k.

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Examples

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Programs

Mathematica

Python

Python

Formula

A346376 a(n) = n^4 + 14n^3 + 63n^2 + 98*n + 28.

A356754 Triangle read by rows: T(n,k) = ((n-1)(n+2))/2 + 2k.