login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A195770 Positive integers a for which there is a 1-Pythagorean triple (a,b,c) satisfying a<=b. 217
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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.

  ...

Related sequences (k-Pythagorean triples):

k...(a(1),b(1),c(1))........a(n).....b(n).....c(n)

0.......(3,4,5).............A009004..A156681..A156682

1.......(3,5,7).............A195770..A195866..A195867

3.......(3,7,11)............A196112..A196113..A196114

4.......(3,8,13)............A196119..A196120..A196121

5.......(1,3,5).............A196155..A196156..A196157

6.......(2,3,7).............A196162..A196163..A196164

7.......(1,1,3).............A196169..A196170..A196171

8.......(1,4,7).............A196176..A196177..A196178

9.......(1,15,19)...........A196183..A196184..A196185

10......(1,2,5).............A196238..A196239..A196240

1/2.....(2,3,4).............A195879..A195880..A195881

3/2.....(5,18,22)...........A195925..A195926..A195927

1/3.....(3,8,9).............A195939..A195940..A195941

2/3.....(4,9,11)............A196001..A196002..A196003

4/3.....(7,36,41)...........A196040..A196041..A196042

5/3.....(7,39,45)...........A196088..A196089..A196090

5/2.....(5,22,28)...........A196026..A196027..A196028

1/4.....(2,2,3).............A196259..A196260..A196261

3/4.....(2,6,7).............A196252..A196253..A196254

5/4.....(3,20,22)...........A196098..A196099..A196100

7/4.....(9,68,76)...........A196105..A196106..A196107

1/5.....(5,7,9).............A196348..A196349..A196350

1/8.....(4,10,11)...........A196355..A196356..A196357

-1......(1,1,1).............A195778..A195794..A195795

-3......(1,3,1).............A196369..A196370..A196371

-4......(1,4,1..............A196376..A196377..A196378

-5......(1,5,1).............A196383..A196384..A196385

-6......(1,6,1).............A196390..A196391..A196392

-1/2....(1,2,2).............A195872..A195873..A195874

-3/2....(2,3,2).............A195918..A195919..A195920

-5/2....(2,5,2).............A196362..A196363..A196364

-1/3....(1,3,3).............A195932..A195933..A195934

-2/3....(2,3,3).............A195994..A195995..A195996

-4/3....(3,4,3).............A196033..A196034..A196035

-5/3....(3,5,3).............A196008..A196009..A196010

-1/4....(1,4,4).............A196266..A196267..A196268

-3/4....(3,4,4).............A196245..A196246..A196247

...

Related sequences (primitive k-Pythagorean triples):

k...(a(1),b(1),c(1))........a(n).....b(n).....c(n)

0.......(3,4,5).............A020884..A156678..A156679

1.......(3,5,7).............A195868..A195869..A195870

3.......(3,7,11)............A196115..A196116..A196117

4.......(3,8,13)............A196122..A196123..A196124

5.......(1,3,5).............A196158..A196159..A196160

6.......(2,3,7).............A196165..A196166..A196167

7.......(1,1,3).............A196172..A196173..A196174

8.......(1,4,7).............A196179..A196180..A196181

9.......(1,15,19)...........A196186..A196187..A196188

10......(1,2,5).............A196241..A196242..A196243

1/2.....(2,3,4).............A195882..A195883..A195884

3/2.....(5,18,22)...........A195928..A195929..A195930

1/3.....(3,8,9).............A195990..A195991..A195992

2/3.....(4,9,11)............A196004..A196005..A196006

4/3.....(7,36,41)...........A196043..A196044..A196045

5/3.....(7,39,45)...........A196091..A196092..A196093

5/2.....(5,22,28)...........A196029..A196030..A196031

1/4.....(2,2,3).............A196262..A196263..A196264

3/4.....(2,6,7).............A196255..A196256..A196257

5/4.....(3,20,22)...........A196101..A196102..A196103

7/4.....(9,68,76)...........A196108..A196109..A196110

1/5.....(5,7,9).............A196351..A196352..A196353

1/8.....(4,10,11)...........A196358..A196359..A196360

-1......(1,1,1))............A195796..A195862..A195863

-3......(1,3,1).............A196372..A196373..A196374

-4......(1,4,1..............A196379..A196380..A196381

-5......(1,5,1).............A196386..A196387..A196388

-6......(1,6,1).............A196393..A196394..A196395

-1/2....(1,2,2).............A195875..A195876..A195877

-3/2....(2,3,2).............A195921..A195922..A195923

-5/2....(2,5,2).............A196365..A196366..A196367

-1/3....(1,3,3).............A195935..A195936..A195937

-2/3....(2,3,3).............A195997..A195998..A195999

-4/3....(3,4,3).............A196036..A196037..A196038

-5/3....(3,5,3).............A196084..A196085..A196086

-1/4....(1,4,4).............A196269..A196270..A196271

-3/4....(3,4,4).............A196248..A196249..A196250

LINKS

Table of n, a(n) for n=1..60.

EXAMPLE

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

MATHEMATICA

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}]  (* A195770 *)

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 *)

CROSSREFS

Cf. A195866, A195867, A195868, A195869, A195870.

Sequence in context: A122818 A070083 A196778 * A196008 A004220 A202308

Adjacent sequences:  A195767 A195768 A195769 * A195771 A195772 A195773

KEYWORD

nonn

AUTHOR

Clark Kimberling, Sep 25 2011

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified March 29 15:10 EDT 2017. Contains 284273 sequences.