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

Offset: 0

Author

Giovanni Corbelli, Jul 26 2021

Keywords

• nonn

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

• 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)

Crossrefs

Cf. A006003, A346637, A346638.

Formula

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

General formula for k-tuples: a_k(n) = n^k - k*binomial(n+1,k) = n^k - (n+1)*binomial(n,k-1).