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

0,3

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).

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

Prog

(Visual Basic) ' See Links.

Crossrefs

Cf. A006003, A346637, A346638.

Sequence in context: A300919 A007834 A228741 * A200873 A082966 A198182

Adjacent sequences: A346633 A346634 A346635 * A346637 A346638 A346639

Keyword

nonn

Author

Giovanni Corbelli, Jul 26 2021

Status

approved