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A380853 Number of ways to place six distinct positive integers on a triangle, three on the corners and three on the sides such that the sum of the three values on each side is n.

%I A380853 #46 Mar 13 2025 06:08:14
%S A380853 0,0,0,0,0,0,0,0,1,3,5,13,14,25,37,47,58,89,98,126,159,188,219,276,
%T A380853 303,362,423,478,536,633,688,781,881,973,1068,1211,1301,1443,1589,
%U A380853 1724,1866,2066,2202,2396,2598,2790,2986,3250,3439,3699,3967,4219,4480,4819,5071
%N A380853 Number of ways to place six distinct positive integers on a triangle, three on the corners and three on the sides such that the sum of the three values on each side is n.
%C A380853 Solutions differing by only rotation or reflections are not counted separately.
%C A380853 If the numbers do not need to be distinct and rotations and reflections are counted separately we get A019298(n-2). If the numbers do not need to be distinct but rotations and reflections do not count separately we get A006918(n-2). If the six numbers must be distinct and reflections and rotations count separately we get 6*a(n). - _R. J. Mathar_, Feb 27 2025
%H A380853 R. J. Mathar, <a href="/A380853/a380853.pdf">Illustrations - Examples</a> (2025)
%H A380853 <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (-1,0,2,3,2,0,-3,-4,-3,0,2,3,2,0,-1,-1).
%F A380853 G.f.: x^9*(1 + 4*x + 8*x^2 + 16*x^3 + 18*x^4 + 18*x^5 + 15*x^6 + 10*x^7)/((1 - x)^4*(1 + 2*x + 2*x^2 + x^3)^2*(1 + x + 2*x^2 + 2*x^3 + 2*x^4 + x^5 + x^6)). - _Stefano Spezia_, Feb 08 2025
%F A380853 A380105(n) = a(n)-a(n-3). - _R. J. Mathar_, Mar 13 2025
%e A380853 The a(9) = 1 solution is:
%e A380853        1
%e A380853      5   6
%e A380853    3   4   2
%p A380853 A380853 := proc(n)
%p A380853     -3412+2353*n+30*n^3-480*n^2+(135*n-900)*(-1)^n ;
%p A380853     %-144*(2*A010891(n)+A010891(n-1)+2*A010891(n-2)) ;
%p A380853     %-160*(17*A049347(n)+8*A049347(n-1)) ;
%p A380853     %-360*A057077(n) ;
%p A380853     %+480*(-1)^n*(A099254(n)-A099254(n-1)) ;
%p A380853     %/720 ;
%p A380853 end proc:
%p A380853 seq(A380853(n),n=1..40) ; # _R. J. Mathar_, Feb 27 2025
%o A380853 (PARI) a(n) = my(c=0, t); for(x=3, n-5, t=n-x; for(y=2, min(x-1, t-1), for(z=1, y-1, if(#Set([x, y, z, t-y, t-z, n-y-z])==6, c++)))); c; \\ _Jinyuan Wang_, Feb 07 2025
%Y A380853 Cf. A342467, A380105.
%K A380853 nonn,easy
%O A380853 1,10
%A A380853 _Derek Holton_ and Alex Holton, Feb 06 2025