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A211802 R(k,n) = number of ordered triples (w,x,y) with all terms in {1,...,n} and 2*w^k < x^k + y^k; square array read by descending antidiagonals.

%I A211802 #21 Jul 06 2024 15:55:05
%S A211802 0,3,0,11,3,0,28,13,3,0,56,32,13,3,0,99,64,34,13,3,0,159,113,68,34,13,
%T A211802 3,0,240,181,117,70,34,13,3,0,344,272,187,125,70,34,13,3,0,475,388,
%U A211802 282,197,125,70,34,13,3,0,635,535,406,292,203,125,70,34,13,3,0
%N A211802 R(k,n) = number of ordered triples (w,x,y) with all terms in {1,...,n} and 2*w^k < x^k + y^k; square array read by descending antidiagonals.
%C A211802 Row 1:  A182260.
%C A211802 Row 2:  A211800.
%C A211802 Row 3:  A211801.
%C A211802 Limiting row sequence: A016061.
%C A211802 Let R be the array in this sequence and let R' be the array in A211805.  Then R(k,n) + R'(k,n) = 3^(n-1).
%C A211802 See the Comments at A211790.
%e A211802 Northwest corner:
%e A211802   0   3  11  28  56  99 159 240
%e A211802   0   3  13  32  64 113 181 272
%e A211802   0   3  13  34  68 117 187 282
%e A211802   0   3  13  34  70 125 197 292
%e A211802   0   3  13  34  70 125 203 302
%t A211802 z = 48;
%t A211802 t[k_, n_] := Module[{s = 0},
%t A211802    (Do[If[2 w^k < x^k + y^k, s = s + 1],
%t A211802        {w, 1, #}, {x, 1, #}, {y, 1, #}] &[n]; s)];
%t A211802 Table[t[1, n], {n, 1, z}]  (* A182260 *)
%t A211802 Table[t[2, n], {n, 1, z}]  (* A211800 *)
%t A211802 Table[t[3, n], {n, 1, z}]  (* A211801 *)
%t A211802 TableForm[Table[t[k, n], {k, 1, 12}, {n, 1, 16}]]
%t A211802 Flatten[Table[t[k, n - k + 1], {n, 1, 12},
%t A211802                 {k, 1, n}]] (* this sequence *)
%t A211802 Table[k (k - 1) (4 k + 1)/6, {k, 1,
%t A211802   z}] (* row-limit sequence, A016061 *)
%t A211802 (* _Peter J. C. Moses_, Apr 13 2012 *)
%Y A211802 Cf. A016061, A182260, A211790, A211800, A211801, A211802, A211805.
%K A211802 nonn,tabl
%O A211802 1,2
%A A211802 _Clark Kimberling_, Apr 22 2012
%E A211802 Definition corrected by _Georg Fischer_, Sep 10 2022