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A057886
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Number of integer 4-tuples that give the lengths of the sides of a nondegenerate quadrilateral with perimeter n.
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5
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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
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OFFSET
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1,6
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LINKS
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FORMULA
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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.
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
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EXAMPLE
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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.
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MAPLE
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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
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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