A070083 - OEIS

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A070083

Perimeters of integer triangles, sorted by perimeter, sides lexicographically ordered.

3, 5, 6, 7, 7, 8, 9, 9, 9, 10, 10, 11, 11, 11, 11, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 19, 19, 19, 19, 19

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OFFSET

1,1

COMMENTS

A005044(p) is the number of all integer triangles having perimeter p.

LINKS

Table of n, a(n) for n=1..63.

R. Zumkeller, Integer-sided triangles

FORMULA

a(n) = A070080(n) + A070081(n) + A070082(n).

MATHEMATICA

maxPer = 19; maxSide = Floor[(maxPer-1)/2]; order[{a_, b_, c_}] := (a+b+c)*maxPer^3 + a*maxPer^2 + b*maxPer + c; triangles = Reap[Do[If[ a+b+c <= maxPer && c-b < a < c+b && b-a < c < b+a && c-a < b < c+a, Sow[{a, b, c}]], {a, 1, maxSide}, {b, a, maxSide}, {c, b, maxSide}]][[2, 1]]; Total /@ Sort[triangles, order[#1] < order[#2] &] (* Jean-François Alcover, Jun 12 2012 *)

maxPer = m = 22; 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]]&]; Total /@ triangles (* Jean-François Alcover, Jul 09 2017 *)

CROSSREFS

Sequence in context: A305443 A362074 A122818 * A316851 A358470 A196778

Adjacent sequences: A070080 A070081 A070082 * A070084 A070085 A070086

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, May 05 2002

STATUS

approved