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%I #29 Oct 28 2023 11:55:12
%S 1,7,1,23,7,1,54,22,7,1,105,51,22,7,1,181,97,50,22,7,1,287,166,96,50,
%T 22,7,1,428,263,163,95,50,22,7,1,609,391,255,161,95,50,22,7,1,835,554,
%U 378,253,161,95,50,22,7,1,1111,756,534,374,252,161,95,50,22,7
%N 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 ...
%C Let R be the array in A211790 and let R' be the array in A211793. Then R(k,n) + R'(k,n) = 3^(n-1). Moreover, (row k of R) =(row k of A211796) for k>2, by Fermat's last theorem; likewise, (row k of R')=(row k of A211799) for k>2.
%C ...
%C Generalizations: Suppose that b,c,d are nonzero integers, and let U(k,n) be the number of ordered triples (w,x,y) with all terms in {1,...,n} and b*w*k <R> c*x^k+d*y^k, where the relation <R> is one of these: <, >=, <=, >. What additional assumptions force the limiting row sequence to be essentially one of these: A002412, A000330, A016061, A174723, A051925?
%C In the following guide to related arrays and sequences, U(k,n) denotes the number of (w,x,y) as described in the preceding paragraph:
%C first 3 rows limiting row sequence
%C A211790: w^k < x^k+y^k A004068, A211635, A211650 A002412
%C A211793: w^k >= x^k+y^k A000292, A211636, A211651 A000330
%C A211796: w^k <= x^k+y^k A002413, A211634, A211650 A002412
%C A211799: w^k > x^k+y^k A000292, A211637, A211651 A000330
%C A211802: 2w^k < x^k+y^k A182260, A211800, A211801 A016061
%C A211805: 2w^k >= x^k+y^k A055232, A211803, A211804 A000330
%C A211808: 2w^k <= x^k+y^k A055232, A211806, A211807 A174723
%C A182259: 2w^k > x^k+y^k A182260, A211810, A211811 A051925
%F R(k,n) = n(n-1)(4n+1)/6 for 1<=k<=n, and
%F R(k,n) = Sum{Sum{floor[(x^k+y^k)^(1/k)] : 1<=x<=n, 1<=y<=n}} for 1<=k<=n.
%e Northwest corner:
%e 1, 7, 23, 54, 105, 181, 287, 428, 609
%e 1, 7, 22, 51, 97, 166, 263, 391, 554
%e 1, 7, 22, 50, 96, 163, 255, 378, 534
%e 1, 7, 22, 50, 95, 161, 253, 374, 528
%e 1, 7, 22, 50, 95, 161, 252, 373, 527
%e For n=2 and k>=1, the 7 triples (w,x,y) are (1,1,1), (1,1,2), (1,2,1), (1,2,2), (2,1,2), (2,2,1), (2,2,2).
%t z = 48;
%t t[k_, n_] := Module[{s = 0},
%t (Do[If[w^k < x^k + y^k, s = s + 1],
%t {w, 1, #}, {x, 1, #}, {y, 1, #}] &[n]; s)];
%t Table[t[1, n], {n, 1, z}] (* A004068 *)
%t Table[t[2, n], {n, 1, z}] (* A211635 *)
%t Table[t[3, n], {n, 1, z}] (* A211650 *)
%t TableForm[Table[t[k, n], {k, 1, 12}, {n, 1, 16}]]
%t Flatten[Table[t[k, n - k + 1], {n, 1, 12}, {k, 1, n}]] (* A211790 *)
%t Table[n (n + 1) (4 n - 1)/6,
%t {n, 1, z}] (* row-limit sequence, A002412 *)
%t (* _Peter J. C. Moses_, Apr 13 2012 *)
%Y Cf. A004068 (row1), A211635 (row2), A211650 (row3), A002412.
%Y Cf. A211793, A211796, A211799, A211802, A211805, A211808, A182259
%K nonn,tabl
%O 1,2
%A _Clark Kimberling_, Apr 21 2012
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