A211422 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w^2 + x*y = 0.
1, 9, 17, 25, 41, 49, 57, 65, 81, 105, 113, 121, 137, 145, 153, 161, 193, 201, 225, 233, 249, 257, 265, 273, 289, 329, 337, 361, 377, 385, 393, 401, 433, 441, 449, 457, 505, 513, 521, 529, 545, 553, 561, 569, 585, 609, 617, 625, 657, 713, 753, 761
Offset: 0
Keywords
A211790
Rectangular array: R(k,n) = number of ordered triples (w,x,y) with all terms in {1,...,n} and w^k
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
Comments
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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 c*x^k+d*y^k, where the relation 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
Examples
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).
Crossrefs
Programs
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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 *)
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.
A211636 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w^2>=x^2+y^2.
0, 0, 1, 5, 13, 28, 50, 80, 121, 175, 244, 327, 425, 544, 683, 845, 1028, 1236, 1470, 1733, 2027, 2349, 2706, 3096, 3520, 3985, 4489, 5034, 5619, 6247, 6922, 7641, 8411, 9230, 10102, 11030, 12007, 13043, 14133, 15288, 16504, 17778, 19117
Offset: 0
Keywords
Comments
For a guide to related sequences, see A211422.
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..10000
Programs
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Mathematica
t = Compile[{{n, _Integer}}, Module[{s = 0}, (Do[If[w^2 >= x^2 + y^2, s = s + 1], {w, 1, #}, {x, 1, #}, {y, 1, #}] &[n]; s)]]; Map[t[#] &, Range[0, 50]] (* A211636 *) (* Peter J. C. Moses, Apr 13 2012 *) -
PARI
a(n) = sum(j=0, n, sum(k=1, j-1, sqrtint(j^2-k^2))); \\ Seiichi Manyama, Oct 28 2023
A211799
Rectangular array: R(k,n) = number of ordered triples (w,x,y) with all terms in {1,...,n} and w^k<=x^k+y
0, 0, 0, 1, 1, 0, 4, 5, 1, 0, 10, 13, 5, 1, 0, 20, 26, 14, 5, 1, 0, 35, 48, 29, 14, 5, 1, 0, 56, 78, 53, 30, 14, 5, 1, 0, 84, 119, 88, 55, 30, 14, 5, 1, 0, 120, 173, 134, 90, 55, 30, 14, 5, 1, 0, 165, 240, 195, 138, 91, 55, 30, 14, 5, 1, 0, 220, 323, 270, 201, 139, 91
Offset: 1
Comments
Examples
Northwest corner: 0...0...1...4....10...20...35...56 0...1...5...13...26...48...78...119 0...1...5...14...29...53...88...134 0...1...5...14...30...55...90...138 0...1...5...14...30...55...91...139
Crossrefs
Cf. A211790.
Programs
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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}] (* A000292 *) Table[t[2, n], {n, 1, z}] (* A211637 *) Table[t[3, n], {n, 1, z}] (* A211651 *) TableForm[Table[t[k, n], {k, 1, 12}, {n, 1, 16}]] Flatten[Table[t[k, n - k + 1], {n, 1, 12}, {k, 1, n}]] (* A211799 *) Table[k (k - 1) (2 k - 1)/6, {k, 1, z}] (* row-limit sequence, A000330 *) (* Peter J. C. Moses, Apr 13 2012 *)
A211793 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.
0, 1, 0, 4, 1, 0, 10, 5, 1, 0, 20, 13, 5, 1, 0, 35, 28, 14, 5, 1, 0, 56, 50, 29, 14, 5, 1, 0, 84, 80, 53, 30, 14, 5, 1, 0, 120, 121, 88, 55, 30, 14, 5, 1, 0, 165, 175, 134, 90, 55, 30, 14, 5, 1, 0, 220, 244, 195, 138, 91, 55, 30, 14, 5, 1, 0, 286, 327, 270, 201, 139
Offset: 1
Comments
Limiting row sequence: A000330.
Examples
Northwest corner: 0, 1, 4, 10, 20, 35, 56, 84 0, 1, 5, 13, 28, 50, 80, 121 0, 1, 5, 14, 29, 53, 88, 134 0, 1, 5, 14, 30, 55, 90, 138 0, 1, 5, 14, 30, 55, 91, 139 0, 1, 5, 14, 30, 55, 91, 140
Programs
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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}] (* A000292 *) Table[t[2, n], {n, 1, z}] (* A211636 *) Table[t[3, n], {n, 1, z}] (* A211651 *) TableForm[Table[t[k, n], {k, 1, 12}, {n, 1, 16}]] Flatten[Table[t[k, n - k + 1], {n, 1, 12}, {k, 1, n}]] (* this sequence *) Table[k (k - 1) (2 k - 1)/6, {k, 1, z}] (* row-limit sequence, A000330 *) (* Peter J. C. Moses, Apr 13 2012 *)
Formula
A211790(k,n) + R(k,n) = 3^(n-1).
Comments
Examples
Links
Crossrefs
Programs
Mathematica
t[n_] := t[n] = Flatten[Table[w^2 + x*y, {w, -n, n}, {x, -n, n}, {y, -n, n}]] c[n_] := Count[t[n], 0] t = Table[c[n], {n, 0, 70}] (* A211422 *) (t - 1)/8 (* A120486 *)