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A059329
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Number of 3 X 3 matrices, with elements from {0,...,n}, having the property that the middle element of each of the eight 3-element horizontal, vertical and diagonal lines equals the average of the two end elements.
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2
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1, 2, 7, 12, 25, 38, 63, 88, 129, 170, 231, 292, 377, 462, 575, 688, 833, 978, 1159, 1340, 1561, 1782, 2047, 2312, 2625, 2938, 3303, 3668, 4089, 4510, 4991, 5472, 6017, 6562, 7175, 7788, 8473, 9158, 9919, 10680, 11521, 12362, 13287, 14212
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OFFSET
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0,2
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COMMENTS
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The bisections of the first differences of {a(n)} give A001844 (the centered triangular numbers n^2+(n-1)^2).
Also, the number of 3 X 3 magic squares with elements from {0,...,n} and with duplicate elements allowed. If [[a,b,c], [d,e,f], [g,h,i]] satisfies the property in the description of this sequence, then [[h,a,f], [c,e,g], [d,i,b]] is a magic square, and conversely. - David Radcliffe, Apr 13 2020
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LINKS
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FORMULA
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even: a(2*n) = (4*n^3 + 6*n^2 + 8*n + 3)/3.
odd: a(2*n-1) = (4*n^3 + 2*n)/3. (End)
a(n) = ((1+n)*(9+3*(-1)^n+4*n+2*n^2))/12.
G.f.: (x^2+1)^2 / ((x-1)^4*(x+1)^2). (End)
E.g.f.: (1/12)*(3*(1 - x)*exp(-x) + (9 + 21*x + 12*x^2 + 2*x^3)*exp(x)). - G. C. Greubel, Jan 07 2017
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MATHEMATICA
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LinearRecurrence[{2, 1, -4, 1, 2, -1}, {1, 2, 7, 12, 25, 38}, 50] (* G. C. Greubel, Jan 07 2017 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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