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A211796 Rectangular array: R(k,n) = number of ordered triples (w,x,y) with all terms in {1,...,n} and w^k<=x^k+y^k.

%I A211796 #9 Dec 04 2016 19:46:28
%S A211796 1,8,1,26,7,1,60,22,7,1,115,51,22,7,1,196,99,50,22,7,1,308,168,96,50,
%T A211796 22,7,1,456,265,163,95,50,22,7,1,645,393,255,161,95,50,22,7,1,880,556,
%U A211796 378,253,161,95,50,22,7,1,1166,760,534,374,252,161,95,50,22,7
%N A211796 Rectangular array:  R(k,n) = number of ordered triples (w,x,y) with all terms in {1,...,n} and w^k<=x^k+y^k.
%C A211796 Row 1:  A002413
%C A211796 Row 2:  A211634
%C A211796 Row 3:  A211650
%C A211796 Limiting row sequence: A002412
%C A211796 Let R be the array in A211796 and let R' be the array in A211799.  Then R(k,n)+R'(k,n)=3^(n-1).
%C A211796 See the Comments at A211790.
%e A211796 Northwest corner:
%e A211796 1...8...26...60...115...196...308
%e A211796 1...7...22...51...99....168...265
%e A211796 1...7...22...50...96....163...255
%e A211796 1...7...22...50...95....161...253
%e A211796 1...7...22...50...95....161...252
%t A211796 z = 48;
%t A211796 t[k_, n_] := Module[{s = 0},
%t A211796    (Do[If[w^k <= x^k + y^k, s = s + 1],
%t A211796        {w, 1, #}, {x, 1, #}, {y, 1, #}] &[n]; s)];
%t A211796 Table[t[1, n], {n, 1, z}]  (* A002413 *)
%t A211796 Table[t[2, n], {n, 1, z}]  (* A211634 *)
%t A211796 Table[t[3, n], {n, 1, z}]  (* A211650 *)
%t A211796 TableForm[Table[t[k, n], {k, 1, 12}, {n, 1, 16}]]
%t A211796 Flatten[Table[t[k, n - k + 1], {n, 1, 12}, {k, 1, n}]] (* A211796 *)
%t A211796 Table[k (k - 1) (2 k - 1)/6, {k, 1,
%t A211796   z}] (* row-limit sequence, A002412 *)
%t A211796 (* _Peter J. C. Moses_, Apr 13 2012 *)
%Y A211796 Cf. A211790.
%K A211796 nonn,tabl
%O A211796 1,2
%A A211796 _Clark Kimberling_, Apr 21 2012