OFFSET
1,6
LINKS
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.
FORMULA
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
EXAMPLE
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.
MAPLE
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
MATHEMATICA
Needs["DiscreteMath`Combinatorica`"]; Table[s=Select[Partitions[n], Length[ # ]==4 && #[[1]]<Total[Rest[ # ]] &]; cnt=0; Do[cnt=cnt+Length[ListNecklaces[4, s[[i]], Dihedral]], {i, Length[s]}]; cnt, {n, 50}] (* T. D. Noe, Oct 24 2006 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
John W. Layman, Sep 19 2000
EXTENSIONS
Corrected by T. D. Noe, Oct 24 2006
STATUS
approved