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A195770 Positive integer a is repeated m times, where m is the number of 1-Pythagorean triples (a,b,c) satisfying a<=b.

%I A195770 #38 Oct 24 2024 01:26:00
%S A195770 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,
%T A195770 20,21,21,21,21,22,22,23,23,24,24,25,25,25,26,26,27,27,27,28,28,29,29,
%U A195770 30,30,30,31,31,32,32,32,33,33,33,33
%N A195770 Positive integer a is repeated m times, where m is the number of 1-Pythagorean triples (a,b,c) satisfying a<=b.
%C A195770 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.
%C A195770 Example: the first five (3/2)-Pythagorean triples are
%C A195770 (5,18,22),(6,11,16),(9,11,71),(10,36,44),(12,22,32);
%C A195770 the first five primitive (3/2)-Pythagorean triples are
%C A195770 (5,18,22),(6,11,16),(9,64,71),(13,138,148),(14,75,86).
%C A195770 ...
%C A195770 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.
%C A195770   ...
%C A195770 Related sequences (k-Pythagorean triples):
%C A195770 k...(a(1),b(1),c(1))........a(n).....b(n).....c(n)
%C A195770 0.......(3,4,5).............A009004..A156681..A156682
%C A195770 1.......(3,5,7).............A195770..A195866..A195867
%C A195770 3.......(3,7,11)............A196112..A196113..A196114
%C A195770 4.......(3,8,13)............A196119..A196120..A196121
%C A195770 5.......(1,3,5).............A196155..A196156..A196157
%C A195770 6.......(2,3,7).............A196162..A196163..A196164
%C A195770 7.......(1,1,3).............A196169..A196170..A196171
%C A195770 8.......(1,4,7).............A196176..A196177..A196178
%C A195770 9.......(1,15,19)...........A196183..A196184..A196185
%C A195770 10......(1,2,5).............A196238..A196239..A196240
%C A195770 1/2.....(2,3,4).............A195879..A195880..A195881
%C A195770 3/2.....(5,18,22)...........A195925..A195926..A195927
%C A195770 1/3.....(3,8,9).............A195939..A195940..A195941
%C A195770 2/3.....(4,9,11)............A196001..A196002..A196003
%C A195770 4/3.....(7,36,41)...........A196040..A196041..A196042
%C A195770 5/3.....(7,39,45)...........A196088..A196089..A196090
%C A195770 5/2.....(5,22,28)...........A196026..A196027..A196028
%C A195770 1/4.....(2,2,3).............A196259..A196260..A196261
%C A195770 3/4.....(2,6,7).............A196252..A196253..A196254
%C A195770 5/4.....(3,20,22)...........A196098..A196099..A196100
%C A195770 7/4.....(9,68,76)...........A196105..A196106..A196107
%C A195770 1/5.....(5,7,9).............A196348..A196349..A196350
%C A195770 1/8.....(4,10,11)...........A196355..A196356..A196357
%C A195770 -1......(1,1,1).............A195778..A195794..A195795
%C A195770 -3......(1,3,1).............A196369..A196370..A196371
%C A195770 -4......(1,4,1).............A196376..A196377..A196378
%C A195770 -5......(1,5,1).............A196383..A196384..A196385
%C A195770 -6......(1,6,1).............A196390..A196391..A196392
%C A195770 -1/2....(1,2,2).............A195872..A195873..A195874
%C A195770 -3/2....(2,3,2).............A195918..A195919..A195920
%C A195770 -5/2....(2,5,2).............A196362..A196363..A196364
%C A195770 -1/3....(1,3,3).............A195932..A195933..A195934
%C A195770 -2/3....(2,3,3).............A195994..A195995..A195996
%C A195770 -4/3....(3,4,3).............A196033..A196034..A196035
%C A195770 -5/3....(3,5,3).............A196008..A196009..A196083
%C A195770 -1/4....(1,4,4).............A196266..A196267..A196268
%C A195770 -3/4....(3,4,4).............A196245..A196247..A196248
%C A195770 ...
%C A195770 Related sequences (primitive k-Pythagorean triples):
%C A195770 k...(a(1),b(1),c(1))........a(n).....b(n).....c(n)
%C A195770 0.......(3,4,5).............A020884..A156678..A156679
%C A195770 1.......(3,5,7).............A195868..A195869..A195870
%C A195770 3.......(3,7,11)............A196115..A196116..A196117
%C A195770 4.......(3,8,13)............A196122..A196123..A196124
%C A195770 5.......(1,3,5).............A196158..A196159..A196160
%C A195770 6.......(2,3,7).............A196165..A196166..A196167
%C A195770 7.......(1,1,3).............A196172..A196173..A196174
%C A195770 8.......(1,4,7).............A196179..A196180..A196181
%C A195770 9.......(1,15,19)...........A196186..A196187..A196188
%C A195770 10......(1,2,5).............A196241..A196242..A196243
%C A195770 1/2.....(2,3,4).............A195882..A195883..A195884
%C A195770 3/2.....(5,18,22)...........A195928..A195929..A195930
%C A195770 1/3.....(3,8,9).............A195990..A195991..A195992
%C A195770 2/3.....(4,9,11)............A196004..A196005..A196006
%C A195770 4/3.....(7,36,41)...........A196043..A196044..A196045
%C A195770 5/3.....(7,39,45)...........A196091..A196092..A196093
%C A195770 5/2.....(5,22,28)...........A196029..A196030..A196031
%C A195770 1/4.....(2,2,3).............A196262..A196263..A196264
%C A195770 3/4.....(2,6,7).............A196255..A196256..A196257
%C A195770 5/4.....(3,20,22)...........A196101..A196102..A196103
%C A195770 7/4.....(9,68,76)...........A196108..A196109..A196110
%C A195770 1/5.....(5,7,9).............A196351..A196352..A196353
%C A195770 1/8.....(4,10,11)...........A196358..A196359..A196360
%C A195770 -1......(1,1,1).............A195796..A195862..A195863
%C A195770 -3......(1,3,1).............A196372..A196373..A196374
%C A195770 -4......(1,4,1).............A196379..A196380..A196381
%C A195770 -5......(1,5,1).............A196386..A196387..A196388
%C A195770 -6......(1,6,1).............A196393..A196394..A196395
%C A195770 -1/2....(1,2,2).............A195875..A195876..A195877
%C A195770 -3/2....(2,3,2).............A195921..A195922..A195923
%C A195770 -5/2....(2,5,2).............A196365..A196366..A196367
%C A195770 -1/3....(1,3,3).............A195935..A195936..A195937
%C A195770 -2/3....(2,3,3).............A195997..A195998..A195999
%C A195770 -4/3....(3,4,3).............A196036..A196037..A196038
%C A195770 -5/3....(3,5,3).............A196084..A196085..A196086
%C A195770 -1/4....(1,4,4).............A196269..A196270..A196271
%C A195770 -3/4....(3,4,4).............A196249..A196250..A196246
%C A195770 From _Georg Fischer_, Oct 26 2020: (Start)
%C A195770 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:
%C A195770         A196112 A196113 A196114 - non-primitive 3-Pythagorean
%C A195770     49:      29    1008    1051
%C A195770         A196241 A196242 A196243 - primitive 10-Pythagorean
%C A195770     31:      13     950    1013
%C A195770 This problem affects 62 of the 74 parameter combinations. (End)
%H A195770 Robert Israel, <a href="/A195770/b195770.txt">Table of n, a(n) for n = 1..10000</a>
%e A195770 The first seven 1-Pythagorean triples (a,b,c), ordered as
%e A195770 described above, are as follows:
%e A195770 3,5,7........7^2 = 3^2 + 5^2 + 3*5
%e A195770 5,16,19.....19^2 = 5^2 + 16^2 + 5*16
%e A195770 6,10,14.....14^2 = 6^2 + 10^2 + 6*10
%e A195770 7,8,13
%e A195770 7,33,37
%e A195770 9,15,21
%e A195770 9,56,61
%e A195770 10,32,38
%p A195770 f:= proc(a) local F,r,u,b;
%p A195770     r:= 3*a^2;
%p A195770     nops(select(proc(t) local b; b:= (r/t - t - 2*a)/4;
%p A195770 (t + r/t) mod 4 = 0 and b::integer and b >= a end proc, numtheory:-divisors(3*a^2)));
%p A195770 end proc:
%p A195770 seq(a$f(a),a=1..100); # _Robert Israel_, Jul 04 2024
%t A195770 z8 = 2000; z9 = 400; z7 = 100;
%t A195770 k = 1; c[a_, b_] := Sqrt[a^2 + b^2 + k*a*b];
%t A195770 d[a_, b_] := If[IntegerQ[c[a, b]], {a, b, c[a, b]}, 0]
%t A195770 t[a_] := Table[d[a, b], {b, a, z8}]
%t A195770 u[n_] := Delete[t[n], Position[t[n], 0]]
%t A195770 Table[u[n], {n, 1, 15}]
%t A195770 t = Table[u[n], {n, 1, z8}];
%t A195770 Flatten[Position[t, {}]]
%t A195770 u = Flatten[Delete[t, Position[t, {}]]];
%t A195770 x[n_] := u[[3 n - 2]];
%t A195770 Table[x[n], {n, 1, z7}]  (* this sequence *)
%t A195770 y[n_] := u[[3 n - 1]];
%t A195770 Table[y[n], {n, 1, z7}]  (* A195866 *)
%t A195770 z[n_] := u[[3 n]];
%t A195770 Table[z[n], {n, 1, z7}]  (* A195867 *)
%t A195770 x1[n_] := If[GCD[x[n], y[n], z[n]] == 1, x[n], 0]
%t A195770 y1[n_] := If[GCD[x[n], y[n], z[n]] == 1, y[n], 0]
%t A195770 z1[n_] := If[GCD[x[n], y[n], z[n]] == 1, z[n], 0]
%t A195770 f = Table[x1[n], {n, 1, z9}];
%t A195770 x2 = Delete[f, Position[f, 0]]  (* A195868 *)
%t A195770 g = Table[y1[n], {n, 1, z9}];
%t A195770 y2 = Delete[g, Position[g, 0]]  (* A195869 *)
%t A195770 h = Table[z1[n], {n, 1, z9}];
%t A195770 z2 = Delete[h, Position[h, 0]]  (* A195870 *)
%Y A195770 Cf. A195866, A195867, A195868, A195869, A195870.
%K A195770 nonn
%O A195770 1,1
%A A195770 _Clark Kimberling_, Sep 25 2011
%E A195770 Name corrected by _Robert Israel_, Jul 04 2024