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

1, 14, 85, 336, 1023, 2610, 5860, 11942, 22555, 40068, 67677, 109578, 171157, 259196, 382096, 550116, 775629, 1073394, 1460845, 1958396, 2589763, 3382302, 4367364, 5580666, 7062679, 8859032, 11020933, 13605606, 16676745, 20304984, 24568384, 29552936, 35353081, 42072246, 49823397, 58729608

Offset: 0

Author

N. J. A. Sloane, Jul 07 2014

Keywords

• nonn
• easy

Comments

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

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]
• Index entries for linear recurrences with constant coefficients, signature (6,-14,14,0,-14,14,-6,1).

Programs

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

Formula

G.f.: (1 + 8*x + 15*x^2 + 8*x^3 + x^4) / ((1 - x)^7*(1 + x)).

From Colin Barker, Jan 11 2017: (Start)

a(n) = (15*(127+(-1)^n) + 6432*n + 8936*n^2 + 6480*n^3 + 2570*n^4 + 528*n^5 + 44*n^6) / 1920.

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.

(End)