A020884 -id:A020884 - cp's OEIS frontend

A156678 Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, gcd (A, B) = 1, A < BA020884(n)).

4, 12, 24, 15, 40, 60, 35, 84, 112, 63, 144, 180, 21, 99, 220, 264, 143, 312, 364, 45, 195, 420, 480, 255, 56, 544, 612, 77, 323, 684, 80, 760, 399, 840, 924, 117, 483, 1012, 1104, 55, 575, 1200, 140, 1300, 165, 675, 1404, 1512, 783, 176, 1624, 1740, 91, 221, 899

Offset: 1

Ant King, Feb 15 2009

The ordered sequence of A values is A020884(n) and the ordered sequence of B values is A020883(n) (allowing repetitions) and A024354(n) (excluding repetitions)

			As the first four primitive Pythagorean triples (ordered by increasing A) are (3,4,5), (5,12,13), (7,24,25) and (8,15,17), then a(1)=4, a(2)=12, a(3)=24 and a(4)=15.

Beiler, Albert H.: Recreations In The Theory Of Numbers, Chapter XIV, The Eternal Triangle, Dover Publications Inc., New York, 1964, pp. 104-134.
Sierpinski, W.; Pythagorean Triangles, Dover Publications, Inc., Mineola, New York, 2003.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Ron Knott, Right-angled Triangles and Pythagoras' Theorem

A020884, A020883, A024354, A156679, A042965, A156680, A156681, A042965.

Cf. A037213.

a156678 n = a156678_list !! (n-1)
a156678_list = f 1 1 where
   f u v | v > uu `div` 2        = f (u + 1) (u + 2)
         | gcd u v > 1 || w == 0 = f u (v + 2)
         | otherwise             = v : f u (v + 2)
         where uu = u ^ 2; w = a037213 (uu + v ^ 2)
-- Reinhard Zumkeller, Nov 09 2012

PrimitivePythagoreanTriplets[n_]:=Module[{t={{3,4,5}},i=4,j=5},While[i

a(n) = A020884(n) + A156680(n).

A156679 Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, gcd (A, B) = 1, A < BA020884(n)).

Original entry on oeis.org

5, 13, 25, 17, 41, 61, 37, 85, 113, 65, 145, 181, 29, 101, 221, 265, 145, 313, 365, 53, 197, 421, 481, 257, 65, 545, 613, 85, 325, 685, 89, 761, 401, 841, 925, 125, 485, 1013, 1105, 73, 577, 1201, 149, 1301, 173, 677, 1405, 1513, 785, 185, 1625, 1741, 109, 229

Offset: 1

Ant King, Feb 15 2009

The ordered sequence of A values is A020884(n) and the ordered sequence of C values is A020882(n) (allowing repetitions) and A008846(n) (excluding repetitions).

			As the first four primitive Pythagorean triples (ordered by increasing A) are (3,4,5), (5,12,13), (7,24,25) and (8,15,17), then a(1)=5, a(2)=13, a(3)=25 and a(4)=17.

Beiler, Albert H.: Recreations In The Theory Of Numbers, Chapter XIV, The Eternal Triangle, Dover Publications Inc., New York, 1964, pp. 104-134.
Sierpinski, W.; Pythagorean Triangles, Dover Publications, Inc., Mineola, New York, 2003.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Ron Knott, Right-angled Triangles and Pythagoras' Theorem

Cf. A020884, A020882, A008846, A156678, A156682.

Cf. A037213.

a156679 n = a156679_list !! (n-1)
a156679_list = f 1 1 where
   f u v | v > uu `div` 2        = f (u + 1) (u + 2)
         | gcd u v > 1 || w == 0 = f u (v + 2)
         | otherwise             = w : f u (v + 2)
         where uu = u ^ 2; w = a037213 (uu + v ^ 2)
-- Reinhard Zumkeller, Nov 09 2012

PrimitivePythagoreanTriplets[n_]:=Module[{t={{3,4,5}},i=4,j=5},While[iHarvey P. Dale, May 10 2020 *)

A156680 Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, gcd (A, B) = 1, A < BA020884(n)).

Original entry on oeis.org

1, 7, 17, 7, 31, 49, 23, 71, 97, 47, 127, 161, 1, 79, 199, 241, 119, 287, 337, 17, 167, 391, 449, 223, 23, 511, 577, 41, 287, 647, 41, 721, 359, 799, 881, 73, 439, 967, 1057, 7, 527, 1151, 89, 1249, 113, 623, 1351, 1457, 727, 119, 1567, 1681, 31, 161, 839, 1799, 1921

Offset: 1

Ant King, Feb 15 2009

This sequence contains the differences in the legs of the primitive Pythagorean triples, sorted by shortest side (A020884). If a difference appears once then it must appear infinitely often, for if (m,n) generates a primitive triple with Abs(b-a)=d then so too does (2m+n,m). This corresponds to applying Hall's A matrix, and hence all horizontal lines in the Pythagorean family tree will contain families of primitive triples whose legs differ by the same amount. The sorted differences that can occur are in A058529.

			As the first four primitive Pythagorean triples (ordered by increasing A) are (3,4,5), (5,12,13), (7,24,25) and (8,15,17), then a(1)=4-3=1, a(2)=12-5=7, a(3)=24-7=17 and a(4)=15-8=7.

Barning, F. J. M.; On Pythagorean and quasi-Pythagorean triangles and a generation process with the help of unimodular matrices. (Dutch), Math. Centrum Amsterdam Afd. Zuivere Wisk. ZW-001 (1963).

A. Hall, Genealogy of Pythagorean Triads, The Mathematical Gazette, Vol. 54, No. 390, (December 1970), pp. 377-379.
Ron Knott, Right-angled Triangles and Pythagoras' Theorem

Cf. A020884, A156678, A058529.

PrimitivePythagoreanTriplets[n_]:=Module[{t={{3,4,5}},i=4,j=5},While[i

a(n) = A156678(n) - A020884(n).

A156689 Inradii of primitive Pythagorean triples a^2+b^2=c^2, 0A020884).

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8, 9, 6, 9, 10, 11, 11, 12, 13, 10, 13, 14, 15, 15, 12, 16, 17, 14, 17, 18, 15, 19, 19, 20, 21, 18, 21, 22, 23, 15, 23, 24, 21, 25, 22, 25, 26, 27, 27, 24, 28, 29, 21, 26, 29, 30

Offset: 1

Ant King, Feb 18 2009

The inradius is given by r=1/2 (a+b-c)=ab/(a+b+c)=area/semiperimeter, and the inradii ordered by increasing r are in A020888.

			The eighth primitive Pythagorean triple ordered by increasing a is (13,84,85). As this has inradius 1/2 (13+84-85)=6, we have a(8)=6.

Mohammad K. Azarian, Circumradius and Inradius, Problem S125, Math Horizons, Vol. 15, Issue 4, April 2008, p. 32. Solution published in Vol. 16, Issue 2, November 2008, p. 32.
D. G. Rogers, Putting Pythagoras in the frame, Mathematics Today, The Institute of Mathematics and its Applications, Vol. 44, No. 3, June 2008, pp. 123-125.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Ron Knott, Right-angled Triangles and Pythagoras' Theorem

Cf. A020884, A020888, A156678, A156679.

Cf. A037213.

a156689 n = a156689_list !! (n-1)
a156689_list = f 1 1 where
   f u v | v > uu `div` 2        = f (u + 1) (u + 2)
         | gcd u v > 1 || w == 0 = f u (v + 2)
         | otherwise             = (u + v - w) `div` 2 : f u (v + 2)
         where uu = u ^ 2; w = a037213 (uu + v ^ 2)
-- Reinhard Zumkeller, Nov 09 2012

PrimitivePythagoreanTriplets[n_]:=Module[{t={{3,4,5}},i=4,j=5},While[i

A156689(n)=1/2 (A020884(n)+A156678(n)-A156679(n))

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

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 25 2011

nonn

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..A196083
-1/4....(1,4,4).............A196266..A196267..A196268
-3/4....(3,4,4).............A196245..A196247..A196248
...
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).............A196249..A196250..A196246
From Georg Fischer, Oct 26 2020: (Start)
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:
        A196112 A196113 A196114 - non-primitive 3-Pythagorean
    49:      29    1008    1051
        A196241 A196242 A196243 - primitive 10-Pythagorean
    31:      13     950    1013
This problem affects 62 of the 74 parameter combinations. (End)

			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

Robert Israel, Table of n, a(n) for n = 1..10000

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

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:
seq(a$f(a),a=1..100); # Robert Israel, Jul 04 2024

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 *)

Name corrected by Robert Israel, Jul 04 2024

A042965 Nonnegative integers not congruent to 2 mod 4.

Original entry on oeis.org

0, 1, 3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, 23, 24, 25, 27, 28, 29, 31, 32, 33, 35, 36, 37, 39, 40, 41, 43, 44, 45, 47, 48, 49, 51, 52, 53, 55, 56, 57, 59, 60, 61, 63, 64, 65, 67, 68, 69, 71, 72, 73, 75, 76, 77, 79, 80, 81, 83, 84, 85, 87, 88, 89, 91, 92

Offset: 1

N. J. A. Sloane

Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, (A, B) = 1, A <= B); sequence (starting at 3) gives values of AUB, sorted and duplicates removed. Values of AUBUC give same sequence. - David W. Wilson
These are the nonnegative integers that can be written as a difference of two squares, i.e., n = x^2 - y^2 for integers x,y. - Sharon Sela (sharonsela(AT)hotmail.com), Jan 25 2002. Equivalently, nonnegative numbers represented by the quadratic form x^2-y^2 of discriminant 4. The primes in this sequence are all the odd primes. - N. J. A. Sloane, May 30 2014
Numbers n such that Kronecker(4,n) == mu(gcd(4,n)). - Jon Perry, Sep 17 2002
Count, sieving out numbers of the form 2*(2*n+1) (A016825, "nombres pair-impairs"). A generalized Chebyshev transform of the Jacobsthal numbers: apply the transform g(x) -> (1/(1+x^2)) g(x/(1+x^2)) to the g.f. of A001045(n+2). Partial sums of 1,2,1,1,2,1,.... - Paul Barry, Apr 26 2005
For n>1, equals union of A020883 and A020884. - Lekraj Beedassy, Sep 28 2004
The sequence 1,1,3,4,5,... is the image of A001045(n+1) under the mapping g(x) -> g(x/(1+x^2)). - Paul Barry, Jan 16 2005
With offset 0 starting (1, 3, 4,...) = INVERT transform of A009531 starting (1, 2, -1, -4, 1, 6,...) with offset 0.
Apparently these are the regular numbers modulo 4 [Haukkanan & Toth]. - R. J. Mathar, Oct 07 2011
Numbers of the form x*y in nonnegative integers x,y with x+y even. - Michael Somos, May 18 2013
Convolution of A106510 with A000027. - L. Edson Jeffery, Jan 24 2015
Numbers that are the sum of zero or more consecutive odd positive numbers. - Gionata Neri, Sep 01 2015
Numbers that are congruent to {0, 1, 3} mod 4. - Wesley Ivan Hurt, Jun 10 2016
Nonnegative integers of the form (2+(3*m-2)/4^j)/3, j,m >= 0. - L. Edson Jeffery, Jan 02 2017
This is { x^2 - y^2; x >= y >= 0 }; with the restriction x > y one gets the same set without zero; with the restriction x > 0 (i.e., differences of two nonzero squares) one gets the set without 1. An odd number 2n-1 = n^2 - (n-1)^2, a number 4n = (n+1)^2 - (n-1)^2. - M. F. Hasler, May 08 2018

			G.f. = x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 7*x^6 + 8*x^7 + 9*x^8 + 11*x^9 + 12*x^10 + ...

R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section D9.
J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 83.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Pentti Haukkanen and László Tóth, An analogue of Ramanujan's sum with respect to regular integers (mod r), Ramanujan J., Vol. 27, No. 1 (2012), pp. 71-88.
Ron Knott, Pythagorean Triples and Online Calculators.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references).
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A001045, A006165, A009531, A020883, A020884, A047209, A047222, A214546, A143978, A042968.
Essentially the complement of A016825.
See A267958 for these numbers multiplied by 4.

a042965 =  (`div` 3) . (subtract 3) . (* 4)
a042965_list = 0 : 1 : 3 : map (+ 4) a042965_list
-- Reinhard Zumkeller, Nov 09 2012

[n: n in [0..100] | not n mod 4 in [2]]; // Vincenzo Librandi, Sep 03 2015

a_list := proc(len) local rec; rec := proc(n) option remember;
ifelse(n <= 4, [0, 1, 3, 4][n], rec(n-1) + rec(n-3) - rec(n-4)) end:
seq(rec(n), n=1..len) end: a_list(76); # Peter Luschny, Aug 06 2022

nn=100; Complement[Range[0,nn], Range[2,nn,4]] (* Harvey P. Dale, May 21 2011 *)
f[n_]:=Floor[(4*n-3)/3]; Array[f,70] (* Robert G. Wilson v, Jun 26 2012 *)
LinearRecurrence[{1, 0, 1, -1}, {0, 1, 3, 4}, 70] (* L. Edson Jeffery, Jan 21 2015 *)
Select[Range[0, 100], ! MemberQ[{2}, Mod[#, 4]] &] (* Vincenzo Librandi, Sep 03 2015 *)

a(n)=(4*n-3)\3 \\ Charles R Greathouse IV, Jul 25 2011

def A042965(n): return (n<<2)//3-1 # Chai Wah Wu, Feb 10 2025

Recurrence: a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.
a(n) = n - 1 + (3n-3-sqrt(3)*(1-2*cos(2*Pi*(n-1)/3))*sin(2*Pi*(n-1)/3))/9. Partial sums of the period-3 sequence 0, 1, 1, 2, 1, 1, 2, 1, 1, 2, ... (A101825). - Ralf Stephan, May 19 2013
G.f.: A(x) = x^2*(1+x)^2/((1-x)^2*(1+x+x^2)); a(n)=Sum{k=0..floor(n/2)}, binomial(n-k-1, k)*A001045(n-2*k), n>0. - Paul Barry, Jan 16 2005, R. J. Mathar, Dec 09 2009
a(n) = floor((4*n-3)/3). - Gary Detlefs, May 14 2011
A214546(a(n)) != 0. - Reinhard Zumkeller, Jul 20 2012
From Michael Somos, May 18 2013: (Start)
Euler transform of length 3 sequence [3, -2, 1].
a(2-n) = -a(n). (End)
From Wesley Ivan Hurt, Jun 10 2016: (Start)
a(n) = (12*n-12+3*cos(2*n*Pi/3)+sqrt(3)*sin(2*n*Pi/3))/9.
a(3k) = 4k-1, a(3k-1) = 4k-3, a(3k-2) = 4k-4. (End)
a(n) = round((4*n-4)/3). - Mats Granvik, Sep 24 2016
The g.f. A(x) satisfies (A(x)/x)^2 + A(x)/x = x*B(x)^2, where B(x) is the o.g.f. of A042968. - Peter Bala, Apr 12 2017
Sum_{n>=2} (-1)^n/a(n) = log(sqrt(2)+2)/sqrt(2) - (sqrt(2)-1)*log(2)/4. - Amiram Eldar, Dec 05 2021
From Peter Bala, Aug 03 2022: (Start)
a(n) = a(floor(n/2)) + a(1 + ceiling(n/2)) for n >= 2, with a(2) = 1 and a(3) = 3.
a(2*n) = a(n) + a(n+1); a(2*n+1) = a(n) + a(n+2). Cf. A047222 and A006165. (End)
E.g.f.: (9 + 12*exp(x)*(x - 1) + exp(-x/2)*(3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2)))/9. - Stefano Spezia, Apr 05 2023

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Peter Pein and Ralf Stephan, Jun 17 2007

Typos fixed in Gary Detlefs's formula and in PARI program by Reinhard Zumkeller, Nov 09 2012

A020883 Ordered long legs of primitive Pythagorean triangles.

Original entry on oeis.org

4, 12, 15, 21, 24, 35, 40, 45, 55, 56, 60, 63, 72, 77, 80, 84, 91, 99, 105, 112, 117, 120, 132, 140, 143, 144, 153, 156, 165, 168, 171, 176, 180, 187, 195, 208, 209, 220, 221, 224, 231, 240, 247, 252, 253, 255, 260, 264, 272, 273, 275, 285, 288, 299, 304, 308, 312, 323

Offset: 1

Clark Kimberling

nonn

Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, (A, B) = 1, A < B); sequence gives values of B, sorted.
Any term in this sequence is given by f(m,n) = 2*m*n or g(m,n) = m^2 - n^2 where m and n are any two positive integers, m > 1, n < m, the greatest common divisor of m and n is 1, m and n are not both odd; e.g., f(m,n) = f(2,1) = 2*2*1 = 4. - Agola Kisira Odero, Apr 29 2016
All terms are composite. - Thomas Ordowski, Mar 12 2017
a(1) is the only power of 2. - Torlach Rush, Nov 08 2019
The first term appearing twice is 420 = a(75) = a(76) = A024410(1). - Giovanni Resta, Nov 11 2019
From Bernard Schott, May 05 2021: (Start)
Also, ordered sides a of primitive triples (a, b, c) for integer-sided triangles where side a is the harmonic mean of the 2 other sides b and c, i.e., 2/a = 1/b + 1/c with b < a < c.
Example: a(2) = 12, because the second triple is (12, 10, 15) with side a = 12, satisfying 2/12 = 1/10 + 1/15 and 15-12 < 10 < 15+12.
The first term appearing twice 420 corresponds to triples (420, 310, 651) and (420, 406, 435), the second one is 572 = a(101) = a(102) = A024410(2) and corresponds to triples (572, 407, 962) and (572, 455, 770). The terms that appear more than once in this sequence are in A024410.
For the corresponding primitive triples and miscellaneous properties and references, see A343891. (End)

V. Lespinard & R. Pernet, Trigonométrie, Classe de Mathématiques élémentaires, programme 1962, problème B-337 p. 179, André Desvigne.

Giovanni Resta, Table of n, a(n) for n = 1..10000
Ron Knott, Pythagorean Triples and Online Calculators

Cf. A020882, A020884, A020885, A020886, A024354, A024410.

Triangles with 2/a = 1/b + 1/c: A343891 (triples), A020883 (side a), A343892 (side b), A343893 (side c), A343894 (perimeter).

for a from 4 to 325 do
for b from floor(a/2)+1 to a-1 do
c := a*b/(2*b-a);
if c=floor(c) and igcd(a,b,c)=1 and c-bBernard Schott, May 05 2021

Extended and corrected by David W. Wilson

A020886 Ordered semiperimeters of primitive Pythagorean triangles.

Original entry on oeis.org

6, 15, 20, 28, 35, 42, 45, 63, 66, 72, 77, 88, 91, 99, 104, 110, 117, 120, 130, 143, 153, 156, 165, 170, 187, 190, 195, 204, 209, 210, 221, 228, 231, 238, 247, 255, 266, 272, 273, 276, 285, 299, 304, 322, 323, 325, 336, 342, 345, 350, 357, 368, 378, 391, 399

Offset: 1

Clark Kimberling

nonn

k is in this sequence iff A078926(k) > 0.
Also, ordered sides c of primitive triples (a, b, c) for integer-sided triangles where side a is the harmonic mean of the 2 other sides b and c, i.e., 2/a = 1/b + 1/c with b < a < c (A343893). - Bernard Schott, May 06 2021
a(n) are the ordered radii of inscribed circles in squares, from which the tangents to the circles are cut off by primitive Pythagorean triangles. - Alexander M. Domashenko, Oct 17 2024

Ray Chandler, Table of n, a(n) for n = 1..10000 (duplicates removed by Sean A. Irvine)

Subsequence of A005279.
Cf. A024364, A078926, A020882, A020884, A020885.
Triangles with 2/a = 1/b + 1/c:  A343891 (triples), A020883 (side a), A343892 (side b), A343893 (side c), A343894 (perimeter).

isA020886 := proc(an) local r::integer,s::integer ; for r from floor((an/2)^(1/2)) to floor(an^(1/2)) do for s from r-1 to 1 by -2 do if r*(r+s) = an and gcd(r,s) < 2 then RETURN(true) ; fi ; if r*(r+s) < an then break ; fi ; od ; od : RETURN(false) ; end : for n from 2 to 400 do if isA020886(n) then printf("%d,",n) ; fi ; od ; # R. J. Mathar, Jun 08 2006

A078926[n_] := Sum[Boole[n < d^2 < 2n && CoprimeQ[d, n/d]], {d, Divisors[ n/2^IntegerExponent[n, 2]]}];
Select[Range[1000], A078926[#]>0&] (* Jean-François Alcover, Mar 23 2020 *)

is(n,f=factor(n))=my(P=apply(i->f[i,1]^f[i,2],[2-n%2..#f~]),nn=2*n); forvec(v=vector(#P,i,[0,1]), my(d=prod(i=1,#v,P[i]^v[i]),d2=d^2); if(d2n, return(1))); 0
list(lim)=my(v=List()); forfactored(n=6,lim\1, if(is(n[1],n[2]), listput(v,n[1]))); Vec(v) \\ Charles R Greathouse IV, Feb 03 2023

a(n) = A024364(n)/2.

A024352 Numbers which are the difference of two positive squares, c^2 - b^2 with 1 <= b < c.

Original entry on oeis.org

3, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, 23, 24, 25, 27, 28, 29, 31, 32, 33, 35, 36, 37, 39, 40, 41, 43, 44, 45, 47, 48, 49, 51, 52, 53, 55, 56, 57, 59, 60, 61, 63, 64, 65, 67, 68, 69, 71, 72, 73, 75, 76, 77, 79, 80, 81, 83, 84, 85, 87, 88, 89, 91, 92, 93, 95, 96

Offset: 1

David W. Wilson

These are the solutions to the equation x^2 + xy = n where y mod 2 = 0, y is positive and x is any positive integer. - Andrew S. Plewe, Oct 19 2007
Ordered different terms of A120070 = 3, 8, 5, 15, 12, 7, ... (which contains two 15's, two 40's, and two 48's). Complement: A139544. (See A139491.) - Paul Curtz, Sep 01 2009
A024359(a(n)) > 0. - Reinhard Zumkeller, Nov 09 2012
If a(n) mod 6 = 3, n > 1, then a(n) = c^2 - f(a(n))^2 where f(n) = (floor(4*n/3) - 3 - n)/2. For example, 171 = 30^2 - 27^2 and f(171) = 27. - Gary Detlefs, Jul 15 2014

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Ron Knott, Pythagorean Triples and Online Calculators
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Same as A042965 except for initial terms. - Michael Somos, Jun 08 2000
Different from A020884.
Cf. A009005, A020884, A120070, A139544, A139491.

a024352 n = a024352_list !! (n-1)
a024352_list = 3 : drop 4 a042965_list
-- Reinhard Zumkeller, Nov 09 2012

[3] cat [4 +Floor((4*n-3)/3): n in [2..100]]; // G. C. Greubel, Apr 22 2023

Union[Flatten[Table[Select[Table[b^2 - c^2, {c, b-1}], # < 100 &], {b, 100}]]] (* Robert G. Wilson v, Jun 05 2004 *)
LinearRecurrence[{1,0,1,-1},{3,5,7,8,9},70] (* Harvey P. Dale, Dec 20 2021 *)

is(n)=(n%4!=2 && n>4) || n==3 \\ Charles R Greathouse IV, May 31 2013

def A024352(n): return 3 if n==1 else 3+(n<<2)//3 # Chai Wah Wu, Feb 10 2025

def A024352(n): return 4 + ((4*n-3)//3) - int(n==1)
[A024352(n) for n in range(1,101)] # G. C. Greubel, Apr 22 2023

Consists of all positive integers except 1, 4 and numbers == 2 (mod 4).
a(n) = a(n-3) + 4, n > 4.
G.f.: (3 + 2*x + 2*x^2 - 2*x^3 - x^4)/(1 - x - x^3 + x^4). - Ralf Stephan, before May 13 2008
a(n) = a(n-1) + a(n-3) - a(n-4), for n > 5. - Ant King, Oct 03 2011
a(n) = 4 + floor((4*n-3)/3), n > 1. - Gary Detlefs, Jul 15 2014

Edited by N. J. A. Sloane, Sep 19 2008

A024361 Number of primitive Pythagorean triangles with leg n.

Original entry on oeis.org

0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 2, 1, 1, 0, 1, 2, 2, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 1, 2, 0, 2, 2, 1, 0, 2, 2, 1, 0, 1, 2, 2, 0, 1, 2, 1, 0, 2, 2, 1, 0, 2, 2, 2, 0, 1, 4, 1, 0, 2, 1, 2, 0, 1, 2, 2, 0, 1, 2, 1, 0, 2, 2, 2, 0, 1, 2, 1, 0, 1, 4, 2, 0, 2, 2, 1, 0, 2, 2, 2, 0, 2, 2, 1, 0, 2, 2, 1, 0, 1, 2, 4

Offset: 1

David W. Wilson

nonn

Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, (A, B) = 1, A <= B); sequence gives number of times A or B takes value n.
For n > 1, a(n) = 0 for n == 2 (mod 4) (n in A016825).
From Jianing Song, Apr 23 2019: (Start)
Note that all the primitive Pythagorean triangles are given by A = min{2*u*v, u^2 - v^2}, B = max{2*u*v, u^2 - v^2}, C = u^2 + v^2, where u, v are coprime positive integers, u > v and u - v is odd. As a result:
(a) if n is odd, then a(n) is the number of representations of n to the form n = u^2 - v^2, where u, v are coprime positive integers (note that this guarantees that u - v is odd) and u > v. Let s = u + v, t = u - v, then n = s*t, where s and t are unitary divisors of n and s > t, so the number of representations is A034444(n)/2 if n > 1 and 0 if n = 1;
(b) if n is divisible by 4, then a(n) is the number of representations of n to the form n = 2*u*v, where u, v are coprime positive integers (note that this also guarantees that u - v is odd because n/2 is even) and u > v. So u and v must be unitary divisors of n/2, so the number of representations is A034444(n/2)/2. Since n is divisible by 4, A034444(n/2) = A034444(n) so a(n) = A034444(n)/2.
(c) if n == 2 (mod 4), then n/2 is odd, so n = 2*u*v implies that u and v are both odd, which is not acceptable, so a(n) = 0.
a(n) = 0 if n = 1 or n == 2 (mod 4), otherwise a(n) is a power of 2.
The earliest occurrence of 2^k is 2*A002110(k+1) for k > 0. (End)

			a(12) = 2 because 12 appears twice, in (A,B,C) = (5,12,13) and (12,35,37).

Antti Karttunen, Table of n, a(n) for n = 1..20000
Ron Knott, Pythagorean Triples and Online Calculators
J. S. Myers, R. Schroeppel, S. R. Shannon, N. J. A. Sloane, and P. Zimmermann, Three Cousins of Recaman's Sequence, arXiv:2004:14000 [math.NT], April 2020.
Amitabha Tripathi, On Pythagorean triples containing a fixed integer, Fibonacci Quart. 46/47 (2008/09), no. 4, 331-340.
Eric Weisstein's World of Mathematics, Pythagorean Triple

Cf. A024359, A024360, A024362, A024363, A046079, A020883, A020884, A034444.

Table[If[n == 1 || Mod[n, 4] == 2, 0, 2^(Length[FactorInteger[n]] - 1)], {n, 100}]

A024361(n) = if(1==n||(2==(n%4)),0,2^(omega(n)-1)); \\ (after the Mathematica program) - Antti Karttunen, Nov 10 2018

a(n) = A034444(n)/2 = 2^(A001221(n)-1) if n != 2 (mod 4) and n > 1, a(n) = 0 otherwise. - Jianing Song, Apr 23 2019

a(n) = A024359(n) + A024360(n). - Ray Chandler, Feb 03 2020

Incorrect comment removed by Ant King, Jan 28 2011

More terms from Antti Karttunen, Nov 10 2018

cp's OEIS Frontend

A156678 Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, gcd (A, B) = 1, A < BA020884(n)).

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A156679 Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, gcd (A, B) = 1, A < BA020884(n)).

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A156680 Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, gcd (A, B) = 1, A < BA020884(n)).

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A156689 Inradii of primitive Pythagorean triples a^2+b^2=c^2, 0A020884).

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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.

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A042965 Nonnegative integers not congruent to 2 mod 4.

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