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Number of integer triples (x,y,z) with x,y,z>0 and 4*x+6*y+3*n*z < 12*n.
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%I #4 Nov 23 2018 13:12:12

%S 0,0,4,13,27,47,70,102,135,178,220,275,327,393,454,532,603,692,772,

%T 873,963,1075,1174,1298,1407,1542,1660,1807,1935,2093,2230,2400,2547,

%U 2728,2884,3077,3243,3447,3622,3838,4023

%N Number of integer triples (x,y,z) with x,y,z>0 and 4*x+6*y+3*n*z < 12*n.

%C In the Comtet formula the round() must not just encompass the (21n^2+6(-)^n)/8 but also the term -n(17+(-)^n)/4 to be correct. See the Maple code.

%D L. Comtet, Advanced Combinatorics (Reidel 1974), page 122, Exercise #19(3)

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1,1,-1,-1,1).

%F G.f. -x^2*(4+9*x+10*x^2+11*x^3+5*x^4+3*x^5) / ( (x^2+1)*(1+x)^2*(x-1)^3 ).

%p A321988 := proc(n)

%p 2-n*(17+(-1)^n)/4 +(21*n^2+6*(-1)^n)/8;

%p round(%) ;

%p end proc:

%p seq(A321988(n),n=0..30) ;

%K nonn,easy

%O 0,3

%A _R. J. Mathar_, Nov 23 2018