A005044 -id:A005044 - cp's OEIS frontend

A325686 Number of strict length-3 compositions x + y + z = n satisfying x + y != z and x != y + z.

0, 0, 0, 0, 0, 0, 2, 6, 8, 18, 16, 30, 34, 48, 48, 72, 72, 96, 98, 126, 128, 162, 160, 198, 202, 240, 240, 288, 288, 336, 338, 390, 392, 450, 448, 510, 514, 576, 576, 648, 648, 720, 722, 798, 800, 882, 880, 966, 970, 1056, 1056, 1152, 1152, 1248, 1250, 1350, 1352

Offset: 0

Gus Wiseman, May 13 2019

nonn

A strict composition of n is a finite sequence of distinct positive integers summing to n.
From Kevin O'Bryant, Jun 02 2025: (Start)
Also the number of Sidon sets in {0,1,...,n} with 4 elements that contain both 0 and n.
Also, the number of 3-tuples of positive integers with the 6 numbers x, y, z, x+y, y+z, x+y+z=n all distinct. (End)

			The a(6) = 2 through a(10) = 16 compositions:
  (132)  (124)  (125)  (126)  (127)
  (231)  (142)  (143)  (135)  (136)
         (214)  (152)  (153)  (154)
         (241)  (215)  (162)  (163)
         (412)  (251)  (216)  (172)
         (421)  (341)  (234)  (217)
                (512)  (243)  (253)
                (521)  (261)  (271)
                       (315)  (316)
                       (324)  (352)
                       (342)  (361)
                       (351)  (451)
                       (423)  (613)
                       (432)  (631)
                       (513)  (712)
                       (531)  (721)
                       (612)
                       (621)

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

Column k = 3 of A325677.
Cf. A000079, A001399, A005044, A108917, A124278, A143823, A266223, A275972.
Cf. A325676, A325688, A325689, A325690, A325691.

Table[Length[Cases[Join@@Permutations/@IntegerPartitions[n,{3}],{x_,y_,z_}/;x!=y!=z&&x+y!=z &&x!=y+z]],{n,0,30}]

Conjectures from Colin Barker, May 14 2019: (Start)
G.f.: 2*x^6*(1 + 3*x + 3*x^2 + 5*x^3) / ((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>9. (End)
Above conjecture confirmed for n <= 5000. - Fausto A. C. Cariboni, Feb 17 2022

A008679 Expansion of 1/((1-x^3)*(1-x^4)).

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 3, 2, 3, 3, 3, 3, 3, 3, 4, 3, 3, 4, 4, 3, 4, 4, 4, 4, 4, 4, 5, 4, 4, 5, 5, 4, 5, 5, 5, 5, 5, 5, 6, 5, 5, 6, 6, 5, 6, 6, 6, 6, 6, 6, 7, 6, 6, 7, 7, 6, 7, 7, 7, 7, 7, 7, 8, 7

Offset: 0

N. J. A. Sloane

Number of partitions of n into parts 3 and 4. - Reinhard Zumkeller, Feb 09 2009
Convolution of A112689 (shifted left once) by A033999. - R. J. Mathar, Feb 13 2009
With four 0's prepended and offset 0, a(n) is the number of partitions of n into four parts whose largest three parts are equal. - Wesley Ivan Hurt, Jan 06 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 216
A. V. Kitaev and A. Vartanian, Algebroid Solutions of the Degenerate Third Painlevé Equation for Vanishing Formal Monodromy Parameter, arXiv:2304.05671 [math.CA], 2023. See p. 20.
Index entries for linear recurrences with constant coefficients, signature (0,0,1,1,0,0,-1).

Cf. A005044, A033999, A112689.

a:=[1,0,0,1,1,0,1,1];; for n in [8..90] do a[n]:=a[n-3]+a[n-4]-a[n-7]; od; a; # G. C. Greubel, Sep 09 2019

R:=PowerSeriesRing(Integers(), 90); Coefficients(R!( 1/((1-x^3)*(1-x^4)) )); // G. C. Greubel, Sep 09 2019

seq(coeff(series(1/((1-x^3)*(1-x^4)), x, n+1), x, n), n = 0..90); # G. C. Greubel, Sep 09 2019

LinearRecurrence[{0,0,1,1,0,0,-1}, {1,0,0,1,1,0,1}, 90] (* Vladimir Joseph Stephan Orlovsky, Feb 23 2012 *)
CoefficientList[Series[1/((1-x)^2(1+x)(1+x+x^2)(1+x^2)), {x,0,90}], x] (* Vincenzo Librandi, Jun 11 2013 *)

my(x='x+O('x^90)); Vec(1/((1-x^3)*(1-x^4))) \\ G. C. Greubel, Sep 09 2019

def A008679_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(1/((1-x^3)*(1-x^4))).list()
A008679_list(90) # G. C. Greubel, Sep 09 2019

a(n+12) = a(n) + 1. - Reinhard Zumkeller, Feb 09 2009
G.f.: 1/((1-x)^2*(1+x)*(1+x+x^2)*(1+x^2)). - R. J. Mathar, Feb 13 2009
a(n) = 1 + floor(n/3) + floor(-n/4). - Tani Akinari, Sep 02 2013
E.g.f.: (1/72)*(9*exp(-x)+21*exp(x)+6*exp(x)*x+18*cos(x)+24*exp(-x/2)*cos(sqrt(3)*x/2)-18*sin(x)+8*sqrt(3)*exp(-x/2)*sin(sqrt(3)*x/2)). - Stefano Spezia, Sep 09 2019
a(n) = A005044(n+3) - A005044(n+1). - Yuchun Ji, Oct 10 2020
From Wesley Ivan Hurt, Jan 17 2021: (Start)
a(n) = a(n-3) + a(n-4) - a(n-7).
a(n) = Sum_{k=1..floor((n+4)/4)} Sum_{j=k..floor((n+4-k)/3)} Sum_{i=j..floor((n+4-j-k)/2)} [j = i = n+4-i-k-j], where [ ] is the Iverson bracket. (End)

A024164 Number of integer-sided triangles with sides a,b,c, a

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

From Bernard Schott, Oct 10 2020: (Start)
Equivalently: number of integer-sided triangles whose sides a < b < c are in arithmetic progression with perimeter n.
Equivalently: number of integer-sided triangles such that b = (a+c)/2 with a < c and perimeter n.
All the perimeters are multiple of 3 because each perimeter = 3 * middle side b.
For each perimeter n = 12*k with k>0, there exists one and only one such right integer triangle whose triple is (3k, 4k, 5k).
For the corresponding primitive triples and miscellaneous properties and references, see A336750. (End)

			a(9) = 1 for the smallest such triangle (2, 3, 4).
a(12) = 1 for Pythagorean triple (3, 4, 5).
a(15) = 2 for the two triples (3, 5, 7) and (4, 5, 6).

Wikipedia, Integer Triangle
Index entries for linear recurrences with constant coefficients, signature (0,0,1,0,0,1,0,0,-1).

Cf. A336750 (triples), A336751 (smallest side), A307136 (middle side), A336753 (largest side), A336754 (perimeter), this sequence (number of triangles whose perimeter = n), A336755 (primitive triples), A336756 (primitive perimeters), A336757 (number of primitive triangles with perimeter = n).

Cf. A005044 (number of integer-sided triangles with perimeter = n).

A024164[n_] := If[Mod[n, 3] == 0, Floor[(n - 3)/6], 0]; Array[A024164, 100] (* Wesley Ivan Hurt, Nov 01 2020 *)
LinearRecurrence[{0,0,1,0,0,1,0,0,-1},{0,0,0,0,0,0,0,0,1},120] (* Harvey P. Dale, Jun 03 2021 *)

If n = 3*k, then a(n) = floor((n-3)/6) = A004526((n-3)/3), otherwise, a(3k+1) = a(3k+2) = 0. - Bernard Schott, Oct 10 2020
From Wesley Ivan Hurt, Nov 01 2020: (Start)
G.f.: x^9/((x^3 - 1)^2*(x^3 + 1)).
a(n) = a(n-3) + a(n-6) - a(n-9).
a(n) = (1 - ceiling(n/3) + floor(n/3)) * floor((n-3)/6). (End)
E.g.f.: (18 + (x - 6)*cosh(x) + (x - 3)*sinh(x) - exp(-x/2)*((9 + 3*exp(x) + x)*cos(sqrt(3)*x/2) + sqrt(3)*x*sin(sqrt(3)*x/2)))/18. - Stefano Spezia, Feb 29 2024

A070092 Number of isosceles integer triangles with perimeter n and prime side lengths.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 0, 1, 0, 1, 0, 0, 1, 2, 0, 2, 1, 2, 0, 1, 0, 2, 0, 1, 1, 2, 0, 3, 1, 3, 0, 2, 0, 3, 0, 1, 1, 3, 0, 3, 0, 2, 0, 1, 0, 3, 0, 1, 1, 2, 0, 3, 1, 4, 0, 2, 0, 4, 0, 1, 0, 1, 0, 4, 1, 3, 0, 2, 0, 3, 0, 1, 1, 3, 0, 4, 1, 4, 0

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

			For n=17 there are A005044(17)=8 integer triangles: [1,8,8], [2,7,8], [3,6,8], [3,7,7], [4,5,8], [4,6,7], [5,5,7] and [5,6,6]: four are isosceles: [1<8=8], [3<7=7], [5=5<7] and [5<6=6], but only two of them consist of primes, therefore a(17)=2.

R. Zumkeller, Integer-sided triangles

Cf. A070080, A070081, A070082, A059169, A070100, A070108, A070117.

Table[Sum[Sum[(PrimePi[i] - PrimePi[i - 1]) (PrimePi[k] - PrimePi[k - 1]) (PrimePi[n - i - k] - PrimePi[n - i - k - 1]) (KroneckerDelta[i, k] + KroneckerDelta[i, n - i - k] - KroneckerDelta[k, n - i - k]) Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}] (* Wesley Ivan Hurt, May 14 2019 *)

a(n) = A070088(n) - A070090(n).

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} sign(floor((i+k)/(n-i-k+1))) * ([i = k] + [i = n-i-k] - [k = n-i-k]) * A010051(i) * A010051(k) * A010051(n-i-k), where [] is the Iverson bracket. - Wesley Ivan Hurt, May 14 2019

A070103 Number of obtuse integer triangles with perimeter n and prime side lengths.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 1, 0, 0, 0, 0, 0, 2, 0, 1, 0, 3, 0, 2, 0, 2, 0, 1, 0, 3, 0, 2, 0, 1, 0, 2, 0, 0, 0, 0, 0, 3, 0, 1, 0, 4, 0, 5, 0, 4, 0, 2, 0, 1, 0, 1, 0, 2, 0, 2, 0, 3, 0, 1, 0, 6, 0, 4, 0, 6, 0, 6, 0

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

			For n=11 there are A005044(11)=4 integer triangles: [1,5,5], [2,4,5], [3,3,5] and [3,4,4]; only one of the two obtuses ([2,4,5] and [3,3,5]) consists of primes, therefore a(11)=1.

R. Zumkeller, Integer-sided triangles

Cf. A070080, A070081, A070082, A070093, A070098, A070101, A070088, A070105, A070108, A070129.

Table[Sum[Sum[(PrimePi[i] - PrimePi[i - 1]) (PrimePi[k] - PrimePi[k - 1]) (PrimePi[n - i - k] - PrimePi[n - i - k - 1]) (1 - Sign[Floor[(i^2 + k^2)/(n - i - k)^2]]) Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}] (* Wesley Ivan Hurt, May 13 2019 *)

a(n) = A070093(n) - A070098(n).

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} (1 - sign(floor((i^2 + k^2)/(n-i-k)^2))) * sign(floor((i + k)/(n-i-k+1))) * A010051(i) * A010051(k) * A010051(n-i-k). - Wesley Ivan Hurt, May 13 2019

A308061 Number of integer-sided triangles with perimeter n whose side lengths are squarefree.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 2, 1, 1, 0, 2, 1, 3, 2, 5, 3, 3, 2, 2, 1, 3, 3, 6, 4, 3, 2, 6, 4, 7, 7, 11, 8, 8, 6, 11, 8, 9, 6, 13, 9, 10, 6, 13, 9, 16, 12, 20, 15, 17, 12, 15, 10, 16, 14, 15, 11, 12, 10, 16, 12, 18, 18, 25, 19, 21, 19, 26, 19, 29, 27, 35, 27, 29, 27

Offset: 1

Wesley Ivan Hurt, May 10 2019

Wikipedia, Integer Triangle

Cf. A005044, A008683.

Table[Sum[Sum[MoebiusMu[i]^2*MoebiusMu[k]^2*MoebiusMu[n - i - k]^2*Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} sign(floor((i+k)/(n-i-k+1))) * mu(i)^2 * mu(k)^2 * mu(n-i-k)^2, where mu is the Möbius function (A008683).

A325689 Number of length-3 compositions of n such that no part is the sum of the other two.

Original entry on oeis.org

0, 0, 0, 1, 0, 6, 4, 15, 12, 28, 24, 45, 40, 66, 60, 91, 84, 120, 112, 153, 144, 190, 180, 231, 220, 276, 264, 325, 312, 378, 364, 435, 420, 496, 480, 561, 544, 630, 612, 703, 684, 780, 760, 861, 840, 946, 924, 1035, 1012, 1128, 1104, 1225, 1200, 1326, 1300, 1431

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 15 2022

			The a(3) = 1 through a(8) = 12 compositions (empty columns not shown):
  (111)  (113)  (114)  (115)  (116)
         (122)  (141)  (124)  (125)
         (131)  (222)  (133)  (152)
         (212)  (411)  (142)  (161)
         (221)         (151)  (215)
         (311)         (214)  (233)
                       (223)  (251)
                       (232)  (323)
                       (241)  (332)
                       (313)  (512)
                       (322)  (521)
                       (331)  (611)
                       (412)
                       (421)
                       (511)

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

Cf. A000079, A001399, A005044, A008642, A069905, A124278, A266223.

Cf. A325676, A325688, A325690, A325691, A325694.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],And@@Table[#[[i]]!=Total[Delete[#,i]],{i,3}]&]],{n,0,30}]

Conjectures from Colin Barker, May 16 2019: (Start)
G.f.: x^3*(1 - x + 4*x^2) / ((1 - x)^3*(1 + x)^2) for n>5.
a(n) = -(5 + 3*(-1)^n - 2*n) * (n-2) / 4 for n>0.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
(End)

A334678 Largest possible hypotenuse of a Pythagorean triangle with perimeter A010814(n).

Original entry on oeis.org

5, 10, 13, 15, 17, 20, 25, 26, 29, 30, 34, 37, 41, 40, 45, 50, 52, 53, 61, 58, 65, 65, 65, 65, 68, 75, 73, 82, 85, 80, 85, 85, 85, 89, 91, 90, 101, 100, 95, 97, 113, 111, 109, 122, 123, 115, 125, 125, 130, 130, 145, 130, 145, 136, 135, 143, 150, 149, 145, 145, 146, 164, 170, 155

Offset: 1

Wesley Ivan Hurt, May 08 2020

nonn

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

Wikipedia, Integer Triangle
Index entries related to Pythagorean Triples.

Cf. A005044, A010814, A334607.

A057886 Number of integer 4-tuples that give the lengths of the sides of a nondegenerate quadrilateral with perimeter n.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 5, 7, 9, 13, 16, 22, 25, 34, 38, 50, 54, 70, 75, 95, 100, 125, 131, 161, 167, 203, 210, 252, 259, 308, 316, 372, 380, 444, 453, 525, 534, 615, 625, 715, 725, 825, 836, 946, 957, 1078, 1090, 1222, 1234, 1378, 1391, 1547, 1560, 1729, 1743

Offset: 1

John W. Layman, Sep 19 2000

nonn

			There are five quadrilaterals with perimeter 8, with sides (1,1,3,3), (1,2,2,3), (1,2,3,2), (1,3,1,3) and (2,2,2,2), so a(8)=5.

Felix Huber, Table of n, a(n) for n = 1..1000
James East and Ron Niles, Integer polygons of given perimeter, arXiv:1710.11245 [math.CO], 2017.
T. Jenkyns and E. Muller, Triangular triples from ceilings to floors, Amer. Math. Monthly, 107 (Aug. 2000), 634-639.

The Moebius transform is A057887. Cf. A005044.

Cf. A062890.

A057886 := proc(n) local s1, s2, s3, s4, a; a := 0; if 4 <= n then for s1 to floor(1/4*n) do for s2 from s1 to floor(1/3*n - 1/3*s1) do for s3 from max(s2, floor(1/2*n - s1 - s2) + 1) to floor(1/2*n - 1/2*s1 - 1/2*s2) do s4 := n - s1 - s2 - s3; if s1 < s2 and s2 < s3 and s3 < s4 then a := a + 3; elif s2 = s3 and (s1 = s2 or s3 = s4) then a := a + 1; else a := a + 2; end if; end do; end do; end do; end if; return a; end proc; seq(A057886(n), n = 1 .. 56); # Felix Huber, Mar 13 2024

Needs["DiscreteMath`Combinatorica`"]; Table[s=Select[Partitions[n], Length[ # ]==4 && #[[1]]T. D. Noe, Oct 24 2006 *)

Conjecture: a(1)=0 and, for n>1, a(n)=a(n-1)+d(n-1), where d(n)=floor(n/4)*floor((n-2)/4) if n is even and d(n)=floor((n+1)/4) if n is odd.
Conjectures from Colin Barker, Oct 27 2013: (Start)
  a(n) = ((n-1)*((n-2)*n+18)+6*sin((Pi*n)/2)+18*cos((Pi*n)/2))/96 for n even;
  a(n) = (n^3-7*n+6*sin((Pi*n)/2)+18*cos((Pi*n)/2))/96 for n odd.
  G.f.: x^4*(x^3-x^2+1) / ((x-1)^4*(x+1)^3*(x^2+1)). (End)
Conjecture: a(n) = ( 2*n^3-3*n^2+13*n-18 - 3*(n^2-9*n+6)*(-1)^n + 12*(2+(-1)^n)*(-1)^((2*n+(-1)^n-1)/4) )/192. - Luce ETIENNE, Nov 06 2014

Corrected by T. D. Noe, Oct 24 2006

A070090 Number of scalene integer triangles with perimeter n and prime side lengths.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 2, 0, 1, 0, 2, 0, 2, 0, 2, 0, 0, 0, 4, 0, 2, 0, 1, 0, 2, 0, 1, 0, 0, 0, 4, 0, 1, 0, 3, 0, 4, 0, 3, 0, 1, 0, 3, 0, 2, 0, 1, 0, 3, 0, 4, 0, 1, 0, 6, 0, 4, 0, 5, 0, 6, 0

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

			For n=15 there are A005044(15)=7 integer triangles: [1,7,7], [2,6,7], [3,5,7], [3,6,6], [4,4,7], [4,5,6] and [5,5,5]: three are scalene: [2<6<7], [3<5<7] and [4<5<6], but only one consists of primes, therefore a(15)=1.

R. Zumkeller, Integer-sided triangles

Cf. A070080, A070081, A070082, A070088, A005044, A070092, A070097, A070105, A070114.

Table[Sum[Sum[(PrimePi[i] - PrimePi[i - 1]) (PrimePi[k] - PrimePi[k - 1]) (PrimePi[n - i - k] - PrimePi[n - i - k - 1])*Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k + 1, Floor[(n - k - 1)/2]}], {k, Floor[(n - 1)/3]}], {n, 100}] (* Wesley Ivan Hurt, May 13 2019 *)

a(n) = A070088(n) - A070092(n).

a(n) = Sum_{k=1..floor((n-1)/3)} Sum_{i=k+1..floor((n-k-1)/2)} sign(floor((i+k)/(n-i-k+1))) * A010051(i)A010051(k)%20*%20A010051(n-i-k).%20-%20_Wesley%20Ivan%20Hurt">* A010051(k) * A010051(n-i-k). - _Wesley Ivan Hurt, May 13 2019

cp's OEIS Frontend

A325686 Number of strict length-3 compositions x + y + z = n satisfying x + y != z and x != y + z.

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A008679 Expansion of 1/((1-x^3)*(1-x^4)).

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A024164 Number of integer-sided triangles with sides a,b,c, a

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A070092 Number of isosceles integer triangles with perimeter n and prime side lengths.

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A070103 Number of obtuse integer triangles with perimeter n and prime side lengths.

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A308061 Number of integer-sided triangles with perimeter n whose side lengths are squarefree.

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A325689 Number of length-3 compositions of n such that no part is the sum of the other two.

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Mathematica

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A334678 Largest possible hypotenuse of a Pythagorean triangle with perimeter A010814(n).

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