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A211790
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.
18
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
OFFSET
1,2
COMMENTS
...
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.
...
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:
first 3 rows limiting row sequence
FORMULA
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.
EXAMPLE
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).
MATHEMATICA
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 *)
CROSSREFS
Cf. A004068 (row1), A211635 (row2), A211650 (row3), A002412.
Sequence in context: A264615 A261248 A214686 * A064051 A147385 A147347
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Apr 21 2012
STATUS
approved