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A160455 Number of triangles that can be built from rods with lengths 1,2,...,n by using and concatenating all rods. 2
0, 2, 7, 12, 16, 27, 48, 70, 91, 127, 184, 243, 300, 385, 507, 631, 752, 919, 1141, 1365, 1587, 1875, 2241, 2611, 2977, 3434, 3997, 4563, 5125, 5808, 6627, 7450, 8269, 9241, 10384, 11532, 12675, 14008, 15552, 17101, 18644, 20419, 22447 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,2

COMMENTS

a(n) is the number of triples (a,b,c) with b+c > a >= b >=c > 0 such that three disjoint subsets A,B,C of {1,2,...,n} with respective element sums a,b,c exist.

a(n) is also the number of partitions as counted in A160438 with additional constraint that there are only three parts and these satisfy the triangle inequality.

LINKS

H. v. Eitzen, Table of n, a(n) for n=3..10000

"AI", (Sci.math thread that inspired investigating the sequence)

H. v. Eitzen, How to Build Triangles from Integers

FORMULA

If n<=2, a(n)=0 trivially because three edges need at least three rods.

If n>=4, then a(n) = A005044(n*(n+1)/2), i.e. for n big enough all triangles of suitable perimeter can be obtained.

Conjectures from Colin Barker, May 12 2019: (Start)

G.f.: x^4*(2 - x + 2*x^2 - 3*x^3 + 6*x^4 - 5*x^5 + 8*x^6 - 9*x^7 + 11*x^8 - 11*x^9 + 11*x^10 - 10*x^11 + 10*x^12 - 8*x^13 + 5*x^14 - 3*x^15 + x^16) / ((1 - x)^5*(1 + x^2)^3*(1 + x + x^2)*(1 + x^4)).

a(n) = 4*a(n-1) - 9*a(n-2) + 17*a(n-3) - 27*a(n-4) + 37*a(n-5) - 47*a(n-6) + 55*a(n-7) - 59*a(n-8) + 59*a(n-9) - 55*a(n-10) + 47*a(n-11) - 37*a(n-12) + 27*a(n-13) - 17*a(n-14) + 9*a(n-15) - 4*a(n-16) + a(n-17) for n>20.

(End)

EXAMPLE

For n=3, only one integer-sided triangle with perimeter 1+2+3=6 exists, namely (2,2,2). This cannot be built from rods of length 1,2 and 3. Therefore a(3)=0.

For n=4, two triangles with perimeter 1+2+3+4=10 exist: (4,4,2) and (4,3,3); both can be built from the available rods: (4,1+3,2) and (4,3,1+2). Therefore a(4)=2.

CROSSREFS

A002623 is a similar problem where one rod per edge is to be used.

Sequence in context: A190453 A320900 A215247 * A045929 A277598 A105501

Adjacent sequences:  A160452 A160453 A160454 * A160456 A160457 A160458

KEYWORD

easy,nonn

AUTHOR

Hagen von Eitzen, May 14 2009

STATUS

approved

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Last modified September 21 15:47 EDT 2021. Contains 347598 sequences. (Running on oeis4.)