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A211790
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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.
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18
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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, 22, 7, 1, 428, 263, 163, 95, 50, 22, 7, 1, 609, 391, 255, 161, 95, 50, 22, 7, 1, 835, 554, 378, 253, 161, 95, 50, 22, 7, 1, 1111, 756, 534, 374, 252, 161, 95, 50, 22, 7
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OFFSET
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1,2
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COMMENTS
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Row 1: A004068
Row 2: A211635
Row 3: A211650
Limiting row sequence: A002412
...
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.
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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?
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:
A211790: w^k < x^k+y^k
first 3 rows: A004068, A211635, A211650
limiting row sequence: A002412
A211793: w^k >= x^k+y^k
first 3 rows: A000292, A211636, A211651
limiting row sequence: A000330
A211796: w^k <= x^k+y^k
first 3 rows: A002413, A211634, A211650
limiting row sequence: A002412
A211799: w^k > x^k+y^k
first 3 rows: A000292, A211637, A211651
limiting row sequence: A000330
A211802: 2w^k < x^k+y^k
first 3 rows: A182260, A211800, A211801
limiting row sequence: A016061
A211805: 2w^k >= x^k+y^k
first 3 rows: A055232, A211803, A211804
limiting row sequence: A000330
A211808: 2w^k <= x^k+y^k
first 3 rows: A055232, A211806, A211807
limiting row sequence: A174723
A182259: 2w^k > x^k+y^k
first 3 rows: A182260, A211810, A211811
limiting row sequence: A051925
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LINKS
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Table of n, a(n) for n=1..65.
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FORMULA
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R(k,n)=n(n-1)(4n+1)/6 for 1<=k<=n, and
R(k,n)=Sum{Sum{floor[(x^k+y^k)^(1/k)] : 1<=x<=n, 1<=y<=n}} for 1<=k<=n.
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EXAMPLE
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Northwest corner:
1...7...23...54...105...181...287...428...609
1...7...22...51...97....166...263...391...554
1...7...22...50...96....163...255...378...534
1...7...22...50...95....161...253...374...528
1...7...22...50...95....161...252...373...527
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).
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MATHEMATICA
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z = 48;
t[k_, n_] := Module[{s = 0},
(Do[If[w^k < x^k + y^k, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}] &[n]; s)];
Table[t[1, n], {n, 1, z}] (* A004068 *)
Table[t[2, n], {n, 1, z}] (* A211635 *)
Table[t[3, n], {n, 1, z}] (* A211650 *)
TableForm[Table[t[k, n], {k, 1, 12}, {n, 1, 16}]]
Flatten[Table[t[k, n - k + 1], {n, 1, 12}, {k, 1, n}]] (* A211790 *)
Table[n (n + 1) (4 n - 1)/6,
{n, 1, z}] (* row-limit sequence, A002412 *)
(* Peter J. C. Moses, Apr 13 2012 *)
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CROSSREFS
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Cf. A211793, A211796, A211799, A211802, A211805, A211808, A182259
Sequence in context: A264615 A261248 A214686 * A064051 A147385 A147347
Adjacent sequences: A211787 A211788 A211789 * A211791 A211792 A211793
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KEYWORD
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nonn,tabl
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AUTHOR
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Clark Kimberling, Apr 21 2012
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STATUS
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approved
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