This site is supported by donations to The OEIS Foundation.

"Email this user" was broken Aug 14 to 9am Aug 16. If you sent someone a message in this period, please send it again.

 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 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.