Search: a195862 -id:a195862
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A195770
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Positive integer a is repeated m times, where m is the number of 1-Pythagorean triples (a,b,c) satisfying a<=b.
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+10
217
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3, 5, 6, 7, 7, 9, 9, 10, 11, 11, 12, 13, 13, 14, 14, 15, 15, 15, 16, 17, 17, 18, 18, 19, 19, 20, 21, 21, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 25, 26, 26, 27, 27, 27, 28, 28, 29, 29, 30, 30, 30, 31, 31, 32, 32, 32, 33, 33, 33, 33
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OFFSET
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1,1
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COMMENTS
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In case the number k=-cos(C) is a rational number, the law of cosines, c^2=a^2+b^2+k*a*b, can be regarded as a Diophantine equation having positive integer solutions a,b,c satisfying a<=b. The terms "k-Pythagorean triple" and "primitive k-Pythagorean triple" generalize the classical terms corresponding to the case k=0.
Example: the first five (3/2)-Pythagorean triples are
(5,18,22),(6,11,16),(9,11,71),(10,36,44),(12,22,32);
the first five primitive (3/2)-Pythagorean triples are
(5,18,22),(6,11,16),(9,64,71),(13,138,148),(14,75,86).
...
If |k|>2, there is no triangle with sidelengths a,b,c satisfying c^2=a^2+b^2+k*a*b, but this equation is, nevertheless, a Diophantine equation for rational k.
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Related sequences (k-Pythagorean triples):
k...(a(1),b(1),c(1))........a(n).....b(n).....c(n)
...
Related sequences (primitive k-Pythagorean triples):
k...(a(1),b(1),c(1))........a(n).....b(n).....c(n)
The Mathematica program below has fixed limits (z7, z8, z9). Therefore, it misses higher values of b. For example, the following triples are do not show up in the corresponding sequences:
49: 29 1008 1051
31: 13 950 1013
This problem affects 62 of the 74 parameter combinations. (End)
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LINKS
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EXAMPLE
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The first seven 1-Pythagorean triples (a,b,c), ordered as
described above, are as follows:
3,5,7........7^2 = 3^2 + 5^2 + 3*5
5,16,19.....19^2 = 5^2 + 16^2 + 5*16
6,10,14.....14^2 = 6^2 + 10^2 + 6*10
7,8,13
7,33,37
9,15,21
9,56,61
10,32,38
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MAPLE
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f:= proc(a) local F, r, u, b;
r:= 3*a^2;
nops(select(proc(t) local b; b:= (r/t - t - 2*a)/4;
(t + r/t) mod 4 = 0 and b::integer and b >= a end proc, numtheory:-divisors(3*a^2)));
end proc:
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MATHEMATICA
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z8 = 2000; z9 = 400; z7 = 100;
k = 1; c[a_, b_] := Sqrt[a^2 + b^2 + k*a*b];
d[a_, b_] := If[IntegerQ[c[a, b]], {a, b, c[a, b]}, 0]
t[a_] := Table[d[a, b], {b, a, z8}]
u[n_] := Delete[t[n], Position[t[n], 0]]
Table[u[n], {n, 1, 15}]
t = Table[u[n], {n, 1, z8}];
Flatten[Position[t, {}]]
u = Flatten[Delete[t, Position[t, {}]]];
x[n_] := u[[3 n - 2]];
Table[x[n], {n, 1, z7}] (* this sequence *)
y[n_] := u[[3 n - 1]];
Table[y[n], {n, 1, z7}] (* A195866 *)
z[n_] := u[[3 n]];
Table[z[n], {n, 1, z7}] (* A195867 *)
x1[n_] := If[GCD[x[n], y[n], z[n]] == 1, x[n], 0]
y1[n_] := If[GCD[x[n], y[n], z[n]] == 1, y[n], 0]
z1[n_] := If[GCD[x[n], y[n], z[n]] == 1, z[n], 0]
f = Table[x1[n], {n, 1, z9}];
x2 = Delete[f, Position[f, 0]] (* A195868 *)
g = Table[y1[n], {n, 1, z9}];
y2 = Delete[g, Position[g, 0]] (* A195869 *)
h = Table[z1[n], {n, 1, z9}];
z2 = Delete[h, Position[h, 0]] (* A195870 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A195778
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Positive integers a for which there is a (-1)-Pythagorean triple (a,b,c) satisfying a<=b.
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+10
8
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1, 2, 3, 3, 4, 5, 5, 5, 6, 6, 7, 7, 7, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 24, 24, 24, 25, 25, 25
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OFFSET
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1,2
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COMMENTS
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See A195770 for definitions of k-Pythagorean triple, primitive k-Pythagorean triple, and lists of related sequences.
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LINKS
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EXAMPLE
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First five (-1)-Pythagorean triples (A195778):
(1,1,1), (2,2,2), (3,3,3), (3,8,7), (4,4,4).
First five primitive (-1)-Pythagorean triples (A195796):
(1,1,1), (3,8,7), (5,8,7), (5,21,19), (7,15,13).
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MATHEMATICA
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z8 = 800; z9 = 400; z7 = 100;
k = -1; c[a_, b_] := Sqrt[a^2 + b^2 + k*a*b];
d[a_, b_] := If[IntegerQ[c[a, b]], {a, b, c[a, b]}, 0]
t[a_] := Table[d[a, b], {b, a, z8}]
u[n_] := Delete[t[n], Position[t[n], 0]]
Table[u[n], {n, 1, 15}]
t = Table[u[n], {n, 1, z8}];
Flatten[Position[t, {}]]
u = Flatten[Delete[t, Position[t, {}]]];
x[n_] := u[[3 n - 2]];
Table[x[n], {n, 1, z7}] (* A195778 *)
y[n_] := u[[3 n - 1]];
Table[y[n], {n, 1, z7}] (* A195794 *)
z[n_] := u[[3 n]];
Table[z[n], {n, 1, z7}] (* A195795 *)
x1[n_] := If[GCD[x[n], y[n], z[n]] == 1, x[n], 0]
y1[n_] := If[GCD[x[n], y[n], z[n]] == 1, y[n], 0]
z1[n_] := If[GCD[x[n], y[n], z[n]] == 1, z[n], 0]
f = Table[x1[n], {n, 1, z9}];
x2 = Delete[f, Position[f, 0]] (* A195796 *)
g = Table[y1[n], {n, 1, z9}];
y2 = Delete[g, Position[g, 0]] (* A195862 *)
h = Table[z1[n], {n, 1, z9}];
z2 = Delete[h, Position[h, 0]] (* A195863 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A195796
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Positive integers a for which there is a primitive (-1)-Pythagorean triple (a,b,c) satisfying a<=b.
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+10
3
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1, 3, 5, 5, 7, 7, 8, 9, 11, 11, 13, 13, 15, 16, 16, 17, 17, 19, 19, 21, 23, 23, 24, 24, 25, 25, 27, 29, 29, 31, 31, 32, 32, 33, 35, 35, 37, 39, 40, 40, 40, 41, 43, 45, 47, 48, 49, 51, 53, 55, 55, 56, 56, 56, 56, 57, 63, 64, 64, 65, 69, 72, 75, 77, 80, 80, 85, 85, 87
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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See A195770 for definitions of k-Pythagorean triple, primitive k-Pythagorean triple, and lists of related sequences.
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LINKS
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EXAMPLE
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Primitive (-1)-Pythagorean triples
(c^2=a^2+b^2+k*a*b, where k=-1):
1,1,1
3,8,7
5,8,7
5,21,19
7,15,13
8,40,37
9,15,13
11,65,61
11,35,31
13,96,91
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MATHEMATICA
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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