A005044 -id:A005044 - cp's OEIS frontend

A070112 Numbers n such that [A070080(n), A070081(n), A070082(n)] is a scalene integer triangle.

8, 13, 17, 20, 21, 25, 29, 30, 33, 36, 37, 41, 42, 44, 45, 49, 50, 53, 56, 57, 59, 60, 62, 66, 67, 69, 70, 74, 75, 77, 78, 79, 80, 83, 86, 87, 89, 90, 92, 96, 97, 99, 100, 101, 102, 105, 106, 110, 111, 113, 114, 115, 116, 119, 122

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

			a(17)=50: [A070080(50), A070081(50), A070082(50)]=[4<6<8].

Jean-François Alcover, Table of n, a(n) for n = 1..920
Reinhard Zumkeller, Integer-sided triangles

Cf. A005044, A070113, A070114, A070121, A070122, A070123, A070130, A070131, A070132.

m = 55 (* max perimeter *);
sides[per_] := Select[Reverse /@ IntegerPartitions[per, {3}, Range[ Ceiling[per/2]]], #[[1]] < per/2 && #[[2]] < per/2 && #[[3]] < per/2 &];
triangles = DeleteCases[Table[sides[per], {per, 3, m}], {}] // Flatten[#, 1] & // SortBy[Total[#] m^3 + #[[1]] m^2 + #[[2]] m + #[[1]] &];
Position[triangles, {a_, b_, c_} /; a < b < c] // Flatten (* Jean-François Alcover, Oct 12 2021 *)

A124278 Triangle of the number of nondegenerate k-gons having perimeter n and whose sides are nondecreasing, for k=3..n.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 3, 2, 2, 1, 1, 3, 4, 4, 3, 2, 1, 1, 2, 5, 5, 4, 3, 2, 1, 1, 4, 7, 8, 6, 5, 3, 2, 1, 1, 3, 8, 9, 9, 6, 5, 3, 2, 1, 1, 5, 11, 14, 12, 10, 7, 5, 3, 2, 1, 1, 4, 12, 16, 16, 13, 10, 7, 5, 3, 2, 1, 1, 7, 16, 23, 22, 19, 14, 11, 7, 5, 3, 2, 1, 1, 5, 18, 25, 28, 24, 20, 14, 11, 7, 5, 3, 2, 1, 1

Offset: 3

T. D. Noe, Oct 24 2006

For a k-gon to be nondegenerate, the longest side must be shorter than the sum of the remaining sides.

T(n,k) = number of partitions of n into k parts (k >= 3) in which all parts are less than n/2. Also T(n,k) = number of partitions of 2*n into k parts in which all parts are even and less than n. - L. Edson Jeffery, Mar 19 2012

			For polygons having perimeter 7: 2 triangles, 2 quadrilaterals, 2 pentagons, 1 hexagon and 1 heptagon. The triangle begins
1
0 1
1 1 1
1 1 1 1
2 2 2 1 1
1 3 2 2 1 1

T. D. Noe, Rows n=3..102 of triangle, flattened
G. E. Andrews, P. Paule and A. Riese, MacMahon's Partition Analysis IX: k-Gon Partitions, Bull. Austral. Math. Soc. 64 (2001), 321-329.
James East, Ron Niles, Integer polygons of given perimeter, arXiv:1710.11245 [math.CO], 2017.

Cf. A124287 (similar, but with no restriction on the sides).
Cf. A210249 (gives row sums of this sequence for n >= 3).
Cf. A005044 (k=3), A062890 (k=4), A069906 (k=5), A069907 (k=6), A288253 (k=7), A288254 (k=8), A288255 (k=9), A288256 (k=10).

b:= proc(n, i) option remember; `if`(n=0, [1], `if`(i<1, [],
      zip((x, y)-> x+y, b(n, i-1), `if`(i>n, [],
                    [0, b(n-i, i)[]]), 0)))
    end:
T:= n-> b(n, ceil(n/2)-1)[4..n+1][]:
seq(T(n), n=3..20);  # Alois P. Heinz, Jul 15 2013

Flatten[Table[p=IntegerPartitions[n]; Length[Select[p, Length[ # ]==k && #[[1]] < Total[Rest[ # ]]&]], {n,3,30}, {k,3,n}]]
(* second program: *)
QP = QPochhammer; T[n_, k_] := SeriesCoefficient[x^k*(1/QP[x, x, k] + x^(k - 2)/((x-1)*QP[x^2, x^2, k-1])), {x, 0, n}]; Table[T[n, k], {n, 3, 16}, {k, 3, n}] // Flatten (* Jean-François Alcover, Jan 08 2016 *)

G.f. for column k is x^k/(product_{i=1..k} 1-x^i) - x^(2k-2)/(1-x)/(product_{i=1..k-1} 1-x^(2i)).

A070094 Number of acute integer triangles with perimeter n and relatively prime side lengths.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 2, 1, 3, 1, 2, 2, 5, 2, 5, 3, 3, 4, 6, 3, 6, 4, 7, 6, 10, 4, 10, 7, 8, 7, 10, 7, 14, 8, 12, 8, 17, 10, 17, 12, 13, 14, 20, 12, 21, 14, 18, 16, 25, 15, 23, 18, 22, 20, 30, 16, 32, 21, 29, 23, 32, 21, 38, 27, 33, 26, 43, 25

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

a(n) = A051493(n) - A070102(n) - A070109(n).

			For n=10 there are A005044(10) = 2 integer triangles: [2,4,4] and [3,3,4]; both are acute, but GCD(2,4,4)>1, therefore a(9) = 1.

R. Zumkeller, Integer-sided triangles

Cf. A070080, A070081, A070082, A070093, A070096, A070099, A070084, A070119.

A070098 Number of integer triangles with perimeter n which are acute and isosceles.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 2, 3, 3, 4, 3, 4, 4, 4, 4, 5, 4, 5, 5, 6, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 7, 8, 8, 8, 8, 9, 9, 9, 9, 10, 9, 10, 10, 11, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 12, 13, 13, 13, 13, 14, 14, 14, 14, 15, 14, 15, 15, 16, 15

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

Equivalently, the number of obtuse isosceles integer triangles with base n. - Charlie Marion, Jun 18 2019

			For n=9 there are A005044(9)=3 integer triangles: [1,4,4], [2,3,4] and [3,3,3]; both isosceles are also acute.

Marius A. Burtea, Table of n, a(n) for n = 1..10000
Reinhard Zumkeller, Integer-sided triangles

Cf. A070080, A070081, A070082, A059169, A070099, A070100, A070124.

[Floor(k/2)-Floor(k/(2 + Sqrt(2)))-((k + 1) mod 2): k in [1..76]]; // Marius A. Burtea, Jun 21 2019

a(n) = A070093(n)-A024154(n); a(n) = A059169(n)-A070106(n).
a(n) = floor(n/2) - floor(n/(2 + sqrt(2))) - ((n + 1) mod 2). - David Pasino, Jun 27 2016
a(n) = A004526(n-1) - A183138(n). - R. J. Mathar, May 22 2019

A070102 Number of obtuse integer triangles with perimeter n and relatively prime side lengths.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 1, 3, 2, 3, 2, 5, 3, 6, 2, 8, 5, 9, 5, 9, 6, 11, 6, 14, 9, 14, 9, 17, 11, 19, 12, 19, 15, 23, 13, 27, 18, 26, 16, 32, 20, 33, 21, 34, 26, 40, 23, 42, 29, 42, 29, 50, 32, 53, 35, 48, 41, 58, 37, 64, 45, 60, 42, 71

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

a(n) = A051493(n) - A070094(n) - A070109(n).

			For n=9 there are A005044(9)=3 integer triangles: [1,4,4], [2,3,4] and [3,3,3]; only one of them is obtuse: 2^2+3^2<16=4^2 and GCD(2,3,4)=1, therefore a(9)=1.

R. Zumkeller, Integer-sided triangles

Cf. A070080, A070081, A070082, A070101, A051493, A070104, A070107, A070084, A070128.

A325688 Number of length-3 compositions of n such that every distinct consecutive subsequence has a different sum.

Original entry on oeis.org

0, 0, 0, 1, 0, 4, 5, 12, 12, 25, 24, 40, 41, 60, 60, 85, 84, 112, 113, 144, 144, 181, 180, 220, 221, 264, 264, 313, 312, 364, 365, 420, 420, 481, 480, 544, 545, 612, 612, 685, 684, 760, 761, 840, 840, 925, 924, 1012, 1013, 1104, 1104, 1201, 1200, 1300, 1301, 1404

Offset: 0

Gus Wiseman, May 15 2019

nonn

A composition of n is a finite sequence of positive integers summing to n.

Confirmed recurrence relation from Colin Barker for n <= 5000. - Fausto A. C. Cariboni, Feb 13 2022

			The a(3) = 1 through a(8) = 12 compositions:
  (111)  (113)  (114)  (115)  (116)
         (122)  (132)  (124)  (125)
         (221)  (222)  (133)  (143)
         (311)  (231)  (142)  (152)
                (411)  (214)  (215)
                       (223)  (233)
                       (241)  (251)
                       (322)  (332)
                       (331)  (341)
                       (412)  (512)
                       (421)  (521)
                       (511)  (611)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..5000

Column k = 3 of A325687.
Cf. A000217 (all length-3).
Cf. A000079, A001399, A005044, A008642, A069905, A108917, A124278, A266223.
Cf. A325686, A325689, A325690, A325691.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],UnsameQ@@Total/@Union[ReplaceList[#,{_,s__,_}:>{s}]]&]],{n,0,30}]

Conjectures from Colin Barker, May 16 2019: (Start)
G.f.: x^3*(1 + 2*x^2 + 4*x^3 + 5*x^4) / ((1 - x)^3*(1 + x)^2*(1 + x + x^2)).
a(n) = 2*a(n-2) + a(n-3) - a(n-4) - 2*a(n-5) + a(n-7) for n>7.
(End)

A325690 Number of length-3 integer partitions of n whose largest part is not the sum of the other two.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 2, 4, 3, 7, 6, 10, 9, 14, 13, 19, 17, 24, 23, 30, 28, 37, 35, 44, 42, 52, 50, 61, 58, 70, 68, 80, 77, 91, 88, 102, 99, 114, 111, 127, 123, 140, 137, 154, 150, 169, 165, 184, 180, 200, 196, 217, 212, 234, 230, 252, 247, 271, 266, 290, 285, 310

Offset: 0

Gus Wiseman, May 15 2019

nonn

Confirmed recurrence relation from Colin Barker for n <= 10000. - Fausto A. C. Cariboni, Feb 19 2022

			The a(3) = 1 through a(13) = 14 partitions (A = 10, B = 11):
  (111)  (221)  (222)  (322)  (332)  (333)  (433)  (443)  (444)   (544)
         (311)  (411)  (331)  (521)  (432)  (442)  (533)  (543)   (553)
                       (421)  (611)  (441)  (622)  (542)  (552)   (643)
                       (511)         (522)  (631)  (551)  (732)   (652)
                                     (531)  (721)  (632)  (741)   (661)
                                     (621)  (811)  (641)  (822)   (733)
                                     (711)         (722)  (831)   (742)
                                                   (731)  (921)   (751)
                                                   (821)  (A11)   (832)
                                                   (911)          (841)
                                                                  (922)
                                                                  (931)
                                                                  (A21)
                                                                  (B11)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..10000

Column k = 3 of A325592.
Cf. A000041, A001399, A005044, A008642, A069905, A266223.
Cf. A325689, A325691, A325694.

Table[Length[Select[IntegerPartitions[n,{3}],#[[1]]!=#[[2]]+#[[3]]&]],{n,0,30}]

Conjectures from Colin Barker, May 15 2019: (Start)
G.f.: x^3*(1 + x^2 + x^3 + x^4) / ((1 - x)^3*(1 + x)^2*(1 + x^2)*(1 + x + x^2)).
a(n) = a(n-2) + a(n-3) + a(n-4) - a(n-5) - a(n-6) - a(n-7) + a(n-9) for n>8.
(End)

A334607 Number of Pythagorean triangles with perimeter A010814(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 4, 3, 1, 2, 2, 1, 3, 1, 2, 2, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 5, 2, 2, 1, 1, 1, 2, 2, 2, 2, 1, 4, 1, 1, 1

Offset: 1

Wesley Ivan Hurt, May 08 2020

nonn

			a(1) = 1; There is one integer-sided right triangle with perimeter A010814(1) = 12, [3,4,5].
a(2) = 1; There is one integer-sided right triangle with perimeter A010814(2) = 24, [6,8,10].

Wikipedia, Integer Triangle
Index entries related to Pythagorean Triples.

Cf. A005044, A010814.

A107572 List of triples a,b,c that are sidelengths of a scalene triangle; a

Original entry on oeis.org

2, 3, 4, 2, 4, 5, 3, 4, 5, 2, 5, 6, 3, 4, 6, 3, 5, 6, 2, 6, 7, 3, 5, 7, 4, 5, 6, 3, 6, 7, 4, 5, 7, 2, 7, 8, 3, 6, 8, 4, 5, 8, 4, 6, 7, 3, 7, 8, 4, 6, 8, 5, 6, 7, 2, 8, 9, 3, 7, 9, 4, 6, 9, 4, 7, 8, 5, 6, 8, 3, 8, 9, 4, 7, 9, 5, 6, 9, 5, 7, 8, 2, 9, 10, 3, 8, 10, 4, 7, 10, 4, 8, 9, 5, 6, 10, 5, 7, 9, 6, 7, 8, 3

Offset: 1

Clark Kimberling, May 16 2005

The number of such triangles of perimeter n+6 is Alcuin's sequence, as noted at A005044.

			(2,3,4) is the least such triangle, followed by (2,4,5) and then (3,4,5).

Cf. A107573, A107574, A107575, A107576, A107577, A005044, A070080.

A124287 Triangle of the number of integer-sided k-gons having perimeter n, for k=3..n.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 3, 1, 1, 1, 5, 4, 4, 1, 1, 3, 7, 9, 7, 4, 1, 1, 2, 9, 13, 15, 8, 5, 1, 1, 4, 13, 23, 25, 20, 10, 5, 1, 1, 3, 16, 29, 46, 37, 29, 12, 6, 1, 1, 5, 22, 48, 72, 75, 57, 35, 14, 6, 1, 1, 4, 25, 60, 113, 129, 125, 79, 47, 16, 7, 1, 1, 7, 34, 92, 172, 228, 231, 185

Offset: 3

T. D. Noe, Oct 24 2006

Rotations and reversals are counted only once. For a k-gon to be nondegenerate, the longest side must be shorter than the sum of the remaining sides. Column k=3 is A005044, column k=4 is A057886, column k=5 is A124285 and column k=6 is A124286. Note that A124278 counts polygons whose sides are nondecreasing.

			For polygons having perimeter 7: 2 triangles, 3 quadrilaterals, 3 pentagons, 1 hexagon and 1 heptagon. The triangle begins
1
0 1
1 1 1
1 2 1 1
2 3 3 1 1
1 5 4 4 1 1

Andrew Howroyd, Table of n, a(n) for n = 3..1277 (terms 3..212 from T. D. Noe)
James East, Ron Niles, Integer polygons of given perimeter, arXiv:1710.11245 [math.CO], 2017.

Row sums are A293818.

Cf. A293819.

Needs["DiscreteMath`Combinatorica`"]; Table[p=Partitions[n]; Table[s=Select[p,Length[ # ]==k && #[[1]]Jean-François Alcover, Jun 14 2018, after Andrew Howroyd *)

T(n,k)={(sumdiv(gcd(n, k), d, eulerphi(d)*binomial(n/d, k/d))/n + binomial(k\2 + (n-k)\2, k\2) - binomial(n\2, k-1) - binomial(n\4, k\2) - if(k%2, 0, binomial((n+2)\4, k\2)))/2;}
for(n=3, 10, for(k=3, n, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 21 2017

A formula is given in Theorem 1.5 of the East and Niles article.

cp's OEIS Frontend

A070112 Numbers n such that [A070080(n), A070081(n), A070082(n)] is a scalene integer triangle.

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A124278 Triangle of the number of nondegenerate k-gons having perimeter n and whose sides are nondecreasing, for k=3..n.

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A070094 Number of acute integer triangles with perimeter n and relatively prime side lengths.

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A070098 Number of integer triangles with perimeter n which are acute and isosceles.

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Magma

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A070102 Number of obtuse integer triangles with perimeter n and relatively prime side lengths.

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A325688 Number of length-3 compositions of n such that every distinct consecutive subsequence has a different sum.

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Mathematica

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A325690 Number of length-3 integer partitions of n whose largest part is not the sum of the other two.

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Mathematica

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A334607 Number of Pythagorean triangles with perimeter A010814(n).

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A107572 List of triples a,b,c that are sidelengths of a scalene triangle; a

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