cp's OEIS Frontend

A308143 Take all the integer-sided triangles with perimeter n and squarefree sides a, b, and c such that a <= b <= c. a(n) is the sum of all the b's.

%I A308143 #13 May 10 2024 08:51:02
%S A308143 0,0,1,0,2,2,5,3,3,0,8,5,16,11,29,18,18,12,13,7,23,23,51,35,28,20,62,
%T A308143 44,82,79,132,98,100,75,144,108,121,80,185,131,148,87,203,145,265,200,
%U A308143 345,264,300,214,272,187,305,274,301,216,246,210,340,258,406
%N A308143 Take all the integer-sided triangles with perimeter n and squarefree sides a, b, and c such that a <= b <= c. a(n) is the sum of all the b's.
%H A308143 Robert Israel, <a href="/A308143/b308143.txt">Table of n, a(n) for n = 1..1000</a>
%H A308143 Wikipedia, <a href="https://en.wikipedia.org/wiki/Integer_triangle">Integer Triangle</a>
%F A308143 a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} sign(floor((i+k)/(n-i-k+1))) * mu(i)^2 * mu(k)^2 * mu(n-i-k)^2 * i, where mu is the Möbius function (A008683).
%p A308143 f:= proc(n)
%p A308143   local a,b,t;
%p A308143   t:= 0;
%p A308143   for a from 1 to n/3 do
%p A308143     if not a::squarefree then next fi;
%p A308143     for b from max(a, ceil((n+1)/2-a)) to (n-a)/2 do
%p A308143       if b::squarefree and (n-a-b)::squarefree then t:= t+b fi
%p A308143   od od;
%p A308143   t
%p A308143 end proc:
%p A308143 map(f, [$1..100]); # _Robert Israel_, May 09 2024
%t A308143 Table[Sum[Sum[i* MoebiusMu[i]^2*MoebiusMu[k]^2*MoebiusMu[n - k - i]^2 *Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]
%Y A308143 Cf. A008683, A308116.
%K A308143 nonn
%O A308143 1,5
%A A308143 _Wesley Ivan Hurt_, May 14 2019