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A332616 a(n) = value of the cubic form A^3 + B^3 + C^3 - 3ABC evaluated at row n of the table in A331195.

%I A332616 #56 May 27 2025 10:10:00
%S A332616 0,1,2,0,8,9,4,16,5,0,27,28,20,35,18,7,54,28,8,0,64,65,54,72,49,32,91,
%T A332616 56,27,10,128,81,40,11,0,125,126,112,133,104,81,152,108,70,44,189,130,
%U A332616 77,36,13,250,176,108,52,14,0,216,217,200,224,189,160,243
%N A332616 a(n) = value of the cubic form A^3 + B^3 + C^3 - 3ABC evaluated at row n of the table in A331195.
%C A332616 No term in the sequence is congruent to 3 or 6 (mod 9).
%H A332616 Mehmet A. Ates, <a href="/A332616/b332616.txt">Table of n, a(n) for n = 0..19599</a>
%H A332616 Mathematical Association of America, <a href="https://web.archive.org/web/20240416133607/https://www.maa.org/sites/default/files/pdf/Putnam/2019/2019PutnamProblems.pdf">2019 William Lowell Putnam Mathematical Competition Problems</a>
%F A332616 a(n) = A056556(n)^3 + A056557(n)^3 + A056558(n)^3 - 3*A056556(n)*A056557(n)*A056558(n).
%e A332616 For n=3, a(n) = f[1,1,0] = 1^3 + 1^3 + 0^3 - 3*1*1*0 = 2.
%t A332616 SeqSize = 30;
%t A332616 ListSize = 120;
%t A332616 F3List = List[];
%t A332616 f3[a_, b_, c_] := a^3 + b^3 + c^3 - 3*a*b*c
%t A332616 For[i = 0, i <= SeqSize, i++, For[j = 0, j <= i, j++, For[k = 0, k <= j, k++, AppendTo[F3List, f3[i, j, k]]]]]
%t A332616 ListPlot[F3List, PlotLabel -> "a(n)"]
%t A332616 Print["First ", ListSize, " elements of a(n): ", Take[F3List, ListSize]]
%Y A332616 Cf. A056556, A056557, A056558.
%Y A332616 Cf. A330013, A331195.
%Y A332616 Cf. A074232 (in ascending order, strictly positive & without duplicates).
%K A332616 nonn,look
%O A332616 0,3
%A A332616 _Mehmet A. Ates_, Jun 08 2020
%E A332616 Edited by _N. J. A. Sloane_, Aug 06 2020