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A346636 a(n) is the number of quadruples (a_1, a_2, a_3, a_4) having all terms in {1,...,n} such that there exists a quadrilateral with these side lengths.

%I A346636 #33 Mar 02 2024 12:28:19
%S A346636 0,1,16,77,236,565,1156,2121,3592,5721,8680,12661,17876,24557,32956,
%T A346636 43345,56016,71281,89472,110941,136060,165221,198836,237337,281176,
%U A346636 330825,386776,449541,519652,597661,684140,779681,884896,1000417,1126896,1265005,1415436
%N A346636 a(n) is the number of quadruples (a_1, a_2, a_3, a_4) having all terms in {1,...,n} such that there exists a quadrilateral with these side lengths.
%C A346636 The existence of such a four-sided polygon implies that every element of the quadruple is less than the sum of the other elements.
%H A346636 Giovanni Corbelli, <a href="/A346636/a346636.txt">Visual Basic routine for generating number of four-sided polygons</a>
%H A346636 Giovanni Corbelli <a href="https://bit.ly/ProofTuplesPolygons">Proof of the formula: Number of k-tuples with elements in {1,2,…,N} corresponding to k-sided polygons</a>
%H A346636 Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a346/A346636.java">Java program</a> (github)
%F A346636 Formula: a(n) = n^4 - 4*binomial(n+1,4) = n^4 - (n+1)*binomial(n,3).
%F A346636 General formula for k-tuples: a_k(n) = n^k - k*binomial(n+1,k) = n^k - (n+1)*binomial(n,k-1).
%o A346636 (Visual Basic) ' See Links.
%Y A346636 Cf. A006003, A346637, A346638.
%K A346636 nonn
%O A346636 0,3
%A A346636 _Giovanni Corbelli_, Jul 26 2021