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A346638 a(n) is the number of 6-tuples (a_1,a_2,a_3,a_4,a_5,a_6) having all terms in {1,...,n} such that there exists a hexagon with these side-lengths.

%I A346638 #29 Oct 30 2024 19:14:04
%S A346638 0,1,64,729,4096,15619,46614,117481,261640,530181,997228,1766017,
%T A346638 2975688,4808791,7499506,11342577,16702960,24026185,33849432,46813321,
%U A346638 63674416,85318443,112774222,147228313,190040376,242759245,307139716,385160049,479040184,591260671
%N A346638 a(n) is the number of 6-tuples (a_1,a_2,a_3,a_4,a_5,a_6) having all terms in {1,...,n} such that there exists a hexagon with these side-lengths.
%C A346638 The existence of such a six-sided polygon implies that every element of the sextuple is less than the sum of the other elements.
%H A346638 Giovanni Corbelli, <a href="/A346638/a346638.txt">Visual Basic routine for generating number of six-sided polygons</a>
%H A346638 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 A346638 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F A346638 a(n) = n^6 - 6*binomial(n+1,6) = n^6 - (n+1)*binomial(n,5).
%F A346638 General formula for k-tuples: a_k(n) = n^k - k*binomial(n+1,k) = n^k - (n+1)*binomial(n,k-1).
%F A346638 G.f.: x*(1 + 57*x + 302*x^2 + 302*x^3 + 51*x^4 + x^5)/(1 - x)^7. - _Stefano Spezia_, Sep 27 2021
%t A346638 CoefficientList[Series[x (1+57x+302x^2+302x^3+51x^4+x^5)/(1-x)^7,{x,0,40}],x] (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,1,64,729,4096,15619,46614},40] (* _Harvey P. Dale_, Oct 30 2024 *)
%o A346638 (Visual Basic) ' See links.
%Y A346638 Cf. A006003, A346636, A346637.
%K A346638 nonn,easy
%O A346638 0,3
%A A346638 _Giovanni Corbelli_, Jul 26 2021