A346637 a(n) is the number of quintuples (a_1,a_2,a_3,a_4,a_5) having all terms in {1,...,n} such that there exists a pentagon with these side-lengths.

0, 1, 32, 243, 1019, 3095, 7671, 16527, 32138, 57789, 97690, 157091, 242397, 361283, 522809, 737535, 1017636, 1377017, 1831428, 2398579, 3098255, 3952431, 4985387, 6223823, 7696974, 9436725, 11477726, 13857507, 16616593, 19798619, 23450445, 27622271, 32367752

Offset: 0

Author

Giovanni Corbelli, Jul 26 2021

Keywords

• nonn
• easy

Comments

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

Links

• Giovanni Corbelli, Visual Basic routine for generating number of five-sided polygons
• Giovanni Corbelli Proof of the formula: Number of k-tuples with elements in {1,2,...,N} corresponding to k-sided polygons
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A006003, A346636, A346638.

Formula

a(n) = n^5 - 5*binomial(n+1,5) = n^5 - (n+1)*binomial(n,4).

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

G.f.: x*(1 + 26*x + 66*x^2 + 21*x^3 + x^4)/(1 - x)^6. - Stefano Spezia, Sep 27 2021