A143150 A triangle sequence based on the Folium of Descartes: x^3 + y^3 - 3*a*x*y.
-1, 3, 4, 19, 17, 27, 53, 48, 55, 80, 111, 103, 107, 129, 175, 199, 188, 189, 208, 251, 324, 323, 309, 307, 323, 363, 433, 539, 489, 472, 467, 480, 517, 584, 687, 832, 703, 683, 675, 685, 719, 783, 883, 1025, 1215, 971, 948, 937, 944, 975, 1036, 1133, 1272, 1459, 1700
Offset: 1
Examples
Triangle begins as:
-1;
3, 4;
19, 17, 27;
53, 48, 55, 80;
111, 103, 107, 129, 175;
199, 188, 189, 208, 251, 324;
323, 309, 307, 323, 363, 433, 539;
...
References
- Samuel M. Shelby, ed., "CRC Standard Mathematical Tables and Formulae", 12th Edition, Curves and Surfaces (page 421).
Links
- G. C. Greubel, Rows n = 1..100 of triangle, flattened
- Wikipedia, Folium of Descartes
Programs
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GAP
Flat(List([1..10], n-> List([1..n], k-> n^3 -3*n*k +k^3 ))); # G. C. Greubel, Apr 03 2019
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Magma
[[n^3 -3*n*k +k^3: k in [1..n]]: n in [1..10]]; // G. C. Greubel, Apr 03 2019
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Mathematica
Table[n^3 +m^3 -3*n*m, {n,1,10}, {m,1,n}]//Flatten (* modified by G. C. Greubel, Apr 03 2019 *) -
PARI
{T(n,k) = n^3 -3*n*k +k^3}; for(n=1,10, for(k=1,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Apr 03 2019 -
Sage
[[n^3 -3*n*k +n^3 for k in (1..n)] for n in (1..10)] # G. C. Greubel, Apr 03 2019
Formula
T(n,m) = n^3 + m^3 - 3*n*m.
row sums: Sum_{m=1..n} T(n,m) = n^2*(5*n^2 - 4*n - 5)/4. - R. J. Mathar, Oct 24 2008
Comments