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
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Apr 21 2012
STATUS
approved