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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.
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%I #25 Mar 02 2024 12:27:11

%S 0,1,32,243,1019,3095,7671,16527,32138,57789,97690,157091,242397,

%T 361283,522809,737535,1017636,1377017,1831428,2398579,3098255,3952431,

%U 4985387,6223823,7696974,9436725,11477726,13857507,16616593,19798619,23450445,27622271,32367752

%N 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.

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

%H Giovanni Corbelli, <a href="/A346637/a346637.txt">Visual Basic routine for generating number of five-sided polygons</a>

%H 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 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).

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

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

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

%o (Visual Basic) ' See links.

%Y Cf. A006003, A346636, A346638.

%K nonn,easy

%O 0,3

%A _Giovanni Corbelli_, Jul 26 2021