A057887 -id:A057887 - cp's OEIS frontend

A051493 Triangles with perimeter n and relatively prime integer side lengths.

0, 0, 1, 0, 1, 0, 2, 1, 2, 1, 4, 2, 5, 2, 5, 4, 8, 4, 10, 6, 9, 6, 14, 8, 15, 9, 16, 12, 21, 11, 24, 16, 22, 16, 27, 18, 33, 20, 31, 24, 40, 23, 44, 30, 39, 30, 52, 32, 54, 35, 52, 42, 65, 38, 65, 48, 64, 49, 80, 48, 85, 56, 77, 64, 90, 58, 102, 72, 93, 69, 114, 72, 120, 81

Offset: 1

Neil Fernandez

nonn

From Peter Munn, Jul 26 2017: (Start)
The triangles that meet the conditions are listed by nondecreasing n in A070110.
Without the requirement for relatively prime side lengths, this sequence becomes A005044.
Counting the triangles by longest side instead of perimeter, this sequence becomes A123323.
a(n) = A070094(n) + A070102(n) + A070109(n).
(End)

			There are 3 triangles with integer-length sides and perimeter 9: 1-4-4, 2-3-4, 3-3-3. 3-3-3 is omitted because isomorphic to 1-1-1, so a(9)=2.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Transforms

Cf. A005044, A057887, A070110, A123323.

Equivalent sequences, restricted to subsets: A070091 (isosceles), A070094 (acute), A070102 (obtuse), A070109 (right-angled), A070138 (with integer area), A070202 (with integer inradius).

nmax = 100;
A005044[n_] := Quotient[n^2 + 6n Mod[n, 2] + 24, 48];
A = Array[A005044, nmax];
mob[m_, n_] := If[ Mod[m, n] == 0, MoebiusMu[m/n], 0];
Reap[Do[Sow[Sum[mob[n, d] A[[d]], {d, 1, n}]], {n, 1, nmax}]][[2, 1]] (* Jean-François Alcover, Oct 05 2021 *)

Moebius transform of A005044.

Corrected and extended with formula by Christian G. Bower, Nov 15 1999

Formula updated due to change to referenced sequence, and definition clarified by Peter Munn, Jul 26 2017

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

cp's OEIS Frontend

A051493 Triangles with perimeter n and relatively prime integer side lengths.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions