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A244865 Number of 3 X 3 symmetric matrices with row and column sums <= n.

%I A244865 #11 Mar 05 2025 07:42:41
%S A244865 1,14,85,336,1023,2610,5860,11942,22555,40068,67677,109578,171157,
%T A244865 259196,382096,550116,775629,1073394,1460845,1958396,2589763,3382302,
%U A244865 4367364,5580666,7062679,8859032,11020933,13605606,16676745,20304984,24568384,29552936,35353081,42072246,49823397,58729608
%N A244865 Number of 3 X 3 symmetric matrices with row and column sums <= n.
%C A244865 a(n) is the number of perimeter-magic hollow triangles of order 4 (4 elements per edge, 9 elements in total) with magic edge sum n+4. The triangles have 9 elements >=1, not necessarily distinct. Triangles obtained by rotations and flips (D_6 symmetry) are counted as being distinct. Associated triangles of order 3 are counted in A019298. - _R. J. Mathar_, Mar 05 2025
%H A244865 Colin Barker, <a href="/A244865/b244865.txt">Table of n, a(n) for n = 0..1000</a>
%H A244865 R. P. Stanley, <a href="/A002721/a002721.pdf">Examples of Magic Labelings</a>, Unpublished Notes, 1973 [Cached copy, with permission]
%H A244865 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (6,-14,14,0,-14,14,-6,1).
%F A244865 G.f.: (1 + 8*x + 15*x^2 + 8*x^3 + x^4) / ((1 - x)^7*(1 + x)).
%F A244865 From _Colin Barker_, Jan 11 2017: (Start)
%F A244865 a(n) = (15*(127+(-1)^n) + 6432*n + 8936*n^2 + 6480*n^3 + 2570*n^4 + 528*n^5 + 44*n^6) / 1920.
%F A244865 a(n) = 6*a(n-1) - 14*a(n-2) + 14*a(n-3) - 14*a(n-5) + 14*a(n-6) - 6*a(n-7) + a(n-8) for n>7.
%F A244865 (End)
%o A244865 (PARI) Vec((1 + 8*x + 15*x^2 + 8*x^3 + x^4) / ((1 - x)^7*(1 + x)) + O(x^40)) \\ _Colin Barker_, Jan 11 2017
%K A244865 nonn,easy
%O A244865 0,2
%A A244865 _N. J. A. Sloane_, Jul 07 2014