A346636 - OEIS

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.

0, 1, 16, 77, 236, 565, 1156, 2121, 3592, 5721, 8680, 12661, 17876, 24557, 32956, 43345, 56016, 71281, 89472, 110941, 136060, 165221, 198836, 237337, 281176, 330825, 386776, 449541, 519652, 597661, 684140, 779681, 884896, 1000417, 1126896, 1265005, 1415436

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OFFSET

0,3

COMMENTS

The existence of such a four-sided polygon implies that every element of the quadruple is less than the sum of the other elements.

LINKS

Table of n, a(n) for n=0..36.

Giovanni Corbelli, Visual Basic routine for generating number of four-sided polygons

Giovanni Corbelli Proof of the formula: Number of k-tuples with elements in {1,2,…,N} corresponding to k-sided polygons

Sean A. Irvine, Java program (github)

FORMULA

Formula: a(n) = n^4 - 4*binomial(n+1,4) = n^4 - (n+1)*binomial(n,3).