A020888 -id:A020888 - cp's OEIS frontend

A119321 Hypotenuse of primitive Pythagorean triangles sorted on inradius (A020888), then on hypotenuse.

5, 13, 17, 25, 41, 37, 61, 29, 85, 65, 113, 145, 101, 181, 53, 221, 145, 265, 65, 313, 197, 365, 85, 421, 73, 89, 257, 481, 545, 325, 613, 125, 685, 401, 761, 97, 841, 109, 149, 485, 925, 173, 1013, 577, 1105, 185, 1201, 677, 1301, 229, 1405, 785, 1513, 137, 1625

Offset: 1

Lekraj Beedassy, May 14 2006

nonn

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.

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A119322.

Extended by Ray Chandler, Apr 12 2010

A058529 Numbers whose prime factors are all congruent to +1 or -1 modulo 8.

Original entry on oeis.org

1, 7, 17, 23, 31, 41, 47, 49, 71, 73, 79, 89, 97, 103, 113, 119, 127, 137, 151, 161, 167, 191, 193, 199, 217, 223, 233, 239, 241, 257, 263, 271, 281, 287, 289, 311, 313, 329, 337, 343, 353, 359, 367, 383, 391, 401, 409, 431, 433, 439, 449, 457, 463, 479, 487

Offset: 1

William Bagby (bagsbee(AT)aol.com), Dec 24 2000

Numbers of the form x^2 - 2*y^2, where x is odd and x and y are relatively prime. - Franklin T. Adams-Watters, Jun 24 2011
Consider primitive Pythagorean triangles (a^2 + b^2 = c^2, gcd(a, b) = 1, a <= b); sequence gives values b-a, sorted with duplicates removed; terms > 1 in sequence give values of a + b, sorted. (See A046086 and A046087.)
Ordered set of (semiperimeter + radius of largest inscribed circle) of all primitive Pythagorean triangles. Semiperimeter of Pythagorean triangle + radius of largest circle inscribed in triangle = ((a+b+c)/2) + ((a+b-c)/2) = a + b.
The terms of this sequence are all of the form 6*N +- 1, since the prime divisors are, and numbers of this form are closed under multiplication. In fact, all terms are == 1, 7, 17, or 23 (mod 24). - J. T. Harrison (harrison_uk_2000(AT)yahoo.co.uk), Apr 28 2009, edited by Franklin T. Adams-Watters, Jun 24 2011
Is similar to A001132, but includes composites whose factors are in A001132. Can be generated in this manner.
Third side of primitive parallepipeds with square base; that is, integer solution of a^2 + b^2 + c^2 = d^2 with gcd(a,b,c) = 1 and b = c. - Carmine Suriano, May 03 2013
Other than -1, values of difference z-y that solve the Diophantine equation x^2 + y^2 = z^2 + 2. - Carmine Suriano, Jan 05 2015
For k > 1, k is in the sequence iff A330174(k) > 0. - Ray Chandler, Feb 26 2020

B Berggren, Pytagoreiska trianglar. Tidskrift för elementär matematik, fysik och kemi, 17:129-139, 1934.
Olaf Delgado-Friedrichs and Michael O’Keeffe, Edge-transitive lattice nets, Acta Cryst. (2009). A65, 360-363.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
F. Barnes, primitive Pythagorean triangles where a-b is a constant.
Johannes Boot, Draft English translation of B Berggren's (1934) article "Pytagoreiska Trianglar", ResearchGate 2017.
K. S. Brown, Pythagorean graphs.
O. Delgado-Friedrichs and M. O'Keeffe, Edge-transitive lattice nets, Acta Cryst. A, A65 (2009), 360-363.
B. Frénicle, Méthode pour trouver la solution des problèmes par les exclusions, 44 pages (see p. 31). In Divers ouvrages de mathematique .. Par Messieurs de l'Academie Royale des Sciences, in-fol, 6+518+1PP, Paris, 1693. - _Paul Curtz_, Sep 06 2008

Cf. A020882-A020886, A020888, A046086, A046087, A014498, A001132, A001653, A027748, A047522, A330174.

a058529 n = a058529_list !! (n-1)
a058529_list = filter (\x -> all (`elem` (takeWhile (<= x) a001132_list))
                                 $ a027748_row x) [1..]
-- Reinhard Zumkeller, Jan 29 2013

Select[Range[500], Union[Abs[Mod[Transpose[FactorInteger[#]][[1]], 8, -1]]] == {1} &] (* T. D. Noe, Feb 07 2012 *)

is(n)=my(f=factor(n)[,1]%8); for(i=1,#f, if(f[i]!=1 && f[i]!=7, return(0))); 1 \\ Charles R Greathouse IV, Aug 01 2016

a(n) = |A-B|=|j^2-2*k^2|, j=(2*n-1), k,n in N, GCD(j,k)=1, the absolute difference between primitive Pythagorean triple legs (sides adjacent to the right angle). - Roger M Ellingson, Dec 09 2023

More terms from Naohiro Nomoto, Jul 02 2001
Edited by Franklin T. Adams-Watters, Jun 24 2011
Duplicated comment removed and name rewritten by Wolfdieter Lang, Feb 17 2015

A014498 Varying radii of inscribed circles within primitive Pythagorean triples as a function of increasing values of hypotenuse.

Original entry on oeis.org

1, 2, 3, 3, 6, 5, 4, 10, 5, 12, 7, 15, 14, 6, 15, 20, 9, 21, 7, 18, 28, 11, 8, 21, 30, 35, 22, 9, 36, 24, 35, 13, 42, 33, 45, 10, 26, 40, 44, 15, 39, 11, 30, 45, 55, 56, 30, 63, 52, 12, 33, 66, 17, 63, 65, 72, 34, 13, 77, 60, 55, 70, 78, 19, 51, 14, 88, 39, 60, 77, 38, 91, 68, 90

Offset: 1

RALPH PETERSON (ralphp(AT)LIBRARY.NRL.NAVY.MIL)

nonn

Ray Chandler, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pythagorean Triple.
Saltire Software, Pythagorean Triples

For ordered values of (a+b-c)/2 see A020888.

Cf. A046086, A046087, A020882, A087459.

Arrange all primitive Pythagorean triples a, b, c by value of hypotenuse c, then by long leg b; for n-th value of c, sequence gives radius of largest inscribed circle, (a+b-c)/2.
a(n) = (A046086(n) + A046087(n) - A020882(n))/2 = A087459(n)/2.
a(n) = sqrt(A118961(n)*A118962(n)/2). - Lekraj Beedassy, May 07 2006

More terms from Asher Auel May 05 2000

Extended by Ray Chandler, Mar 09 2004

A020887 Ordered set of a + b - c as (a,b,c) runs through all primitive Pythagorean triples with a

Original entry on oeis.org

2, 4, 6, 6, 8, 10, 10, 12, 12, 14, 14, 16, 18, 18, 20, 20, 22, 22, 24, 24, 26, 26, 28, 28, 30, 30, 30, 30, 32, 34, 34, 36, 36, 38, 38, 40, 40, 42, 42, 42, 42, 44, 44, 46, 46, 48, 48, 50, 50, 52, 52, 54, 54, 56, 56, 58, 58, 60, 60, 60, 60, 62, 62, 64, 66, 66, 66, 66, 68, 68, 70

Offset: 1

Clark Kimberling

nonn

Also called "excess" of primitive Pythagorean triangles and corresponds to incircle's diameter. - Lekraj Beedassy, Dec 01 2003

Ray Chandler, Table of n, a(n) for n = 1..10000
D. McCullough, The Height and Excess of Pythagorean Triples

For values ordered by hypotenuse, see A087459.

a(n) = 2*A020888(n).

Offset corrected to 1 by Ray Chandler, Jan 23 2020

A120427 For each y >= 1 there are only finitely many values of x >= 1 such that x-y and x+y are both squares; list all such pairs (x,y) with gcd(x,y) = 1 ordered by values of y; sequence gives y values.

Original entry on oeis.org

4, 8, 12, 12, 16, 20, 20, 24, 24, 28, 28, 32, 36, 36, 40, 40, 44, 44, 48, 48, 52, 52, 56, 56, 60, 60, 60, 60, 64, 68, 68, 72, 72, 76, 76, 80, 80, 84, 84, 84, 84, 88, 88, 92, 92, 96, 96, 100, 100, 104, 104, 108, 108, 112, 112, 116, 116, 120, 120, 120, 120, 124, 124, 128

Offset: 1

N. J. A. Sloane, May 02 2001

nonn

Ordered even legs of primitive Pythagorean triangles.
I wrote an arithmetic program once to find out if and when y 'catches up to' n in A120427 (ordered even legs of primitive Pythagorean triples). It's around 16700. As enumerated by the even - or odd - legs, (not sure about the hypotenuses), the triples are 'denser' than the integers. - Stephen Waldman, Jun 12 2007
Conjecture: lim_{n->oo} a(n)/n = 1/Pi. - Lothar Selle, Jun 19 2022

			Pairs are [5, 4], [17, 8], [13, 12], [37, 12], [65, 16], [29, 20], [101, 20], ... E.g., 5-4 = 1^2, 5+4 = 3^2.

Lothar Selle, Kleines Handbuch Pythagoreische Zahlentripel, Books on Demand, 3rd impression 2022, chapter 2.3.1.
Donald D. Spencer, Computers in Number Theory, Computer Science Press, Rockville MD, 1982, pp. 130-131.

D. N. Lehmer, Asymptotic evaluation of certain totient sums, Amer. J. Math. 22, 293-335, 1900.

Cf. A060829, A061408, A061409.

Even entries of A024355. Ordered union of A081925 and A081935.

The solutions are given by x = r^2 + 2*r*k + 2*k^2, y = 2*k*(k+r) with r >= 1, k >= 1, r odd, gcd(r, k) = 1.

a(n) = 2*A020887(n) = 4*A020888(n).

Corrected by Lekraj Beedassy, Jul 12 2007 and by Stephen Waldman (brogine(AT)gmail.com), Jun 09 2007

A067360 a(n) = 17^n sin(2n arctan(1/4)) or numerator of tan(2n arctan(1/4)).

Original entry on oeis.org

8, 240, 4888, 77280, 905768, 4839120, -116593352, -4896306240, -113193708472, -1980778750800, -26710380775592, -228866364286560, 853309115549288, 91741652745294480, 2505643247965090168, 48655959795562600320, 735547895204966951048

Offset: 1

Barbara Haas Margolius (b.margolius(AT)csuohio.edu), Jan 17 2002

Note that A067360(n), A067361(n) and 17^n are primitive Pythagorean triples with hypotenuse 17^n.

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 430-433.

J. M. Borwein and R. Girgensohn, Addition theorems and binary expansions, Canadian J. Math. 47 (1995) 262-273.
E. Eckert, The group of primitive Pythagorean triangles, Mathematics Magazine 57 (1984) 22-27.
Steven R. Finch, Plouffe's Constant [Broken link]
Steven R. Finch, Plouffe's Constant [From the Wayback machine]
Simon Plouffe, The Computation of Certain Numbers Using a Ruler and Compass, J. Integer Seqs. Vol. 1 (1998), #98.1.3.
Index entries for linear recurrences with constant coefficients, signature (30, -289).

Cf. A067361 (17^n cos(2n arctan(1/4))).

Cf. A066770, A066771, A067358, A067359, A020888, A014498, A020892.

a[1] := 8/15; for n from 1 to 40 do a[n+1] := (8/15+a[n])/(1-8/15*a[n]):od: seq(abs(numer(a[n])), n=1..40);# a[n]=tan(2n arctan(1/4))

Table[Tan[2n ArcTan[1/4]] // TrigToExp // Simplify // Numerator, {n, 1, 17} ] (* Jean-François Alcover, Jul 25 2017 *)

a(n) = 17^n sin(2n arctan(1/4)). A recursive formula for T(n) = tan(2n arctan(1/4)) is T(n+1)=(8/15+T(n))/(1-8/15*T(n)). Unsigned a(n) is the absolute value of numerator of T(n).
Conjectures from Colin Barker, Jul 25 2017: (Start)
G.f.: 8*x / (1 - 30*x + 289*x^2).
a(n) = i*((15 - 8*i)^n - (15 + 8*i)^n)/2 where i=sqrt(-1).
a(n) = 30*a(n-1) - 289*a(n-2) for n>2.
(End)

A067361 a(n) = 17^ncos(2n*arctan(1/4)) or denominator of tan(2narctan(1/4)).

Original entry on oeis.org

15, 161, 495, -31679, -1093425, -23647519, -393425745, -4968639359, -35359140465, 375162560801, 21473668418415, 535788072480961, 9867752001506895, 141189807098209121, 1383913884510780975, 713562283940993281, -378544244105385903345

Offset: 1

Barbara Haas Margolius, (b.margolius(AT)csuohio.edu), Jan 17 2002

Note that A067360(n), A067361(n) and 17^n are primitive Pythagorean triples with hypotenuse 17^n.

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 430-433.

J. M. Borwein and R. Girgensohn, Addition theorems and binary expansions, Canadian J. Math. 47 (1995) 262-273.
E. Eckert, The group of primitive Pythagorean triangles, Mathematics Magazine 57 (1984) 22-27.
Steven R. Finch, Plouffe's Constant [Broken link]
Steven R. Finch, Plouffe's Constant [From the Wayback machine]
Simon Plouffe, The Computation of Certain Numbers Using a Ruler and Compass, J. Integer Seqs. Vol. 1 (1998), #98.1.3.
Index entries for linear recurrences with constant coefficients, signature (30, -289).

Cf. A067360 (17^n sin(2n arctan(1/4))).

Cf. A066770, A066771, A067358, A067359, A020888, A014498, A020892.

a[1] := 8/15; for n from 1 to 40 do a[n+1] := (8/15+a[n])/(1-8/15*a[n]):od: seq(abs(denom(a[n])), n=1..40);# a[n]=tan(2n arctan(1/4))

Table[t = Tan[2 n ArcTan[1/4]] // TrigToExp // Simplify; Sign[t] * Denominator[t], {n, 1, 17}] (* Jean-François Alcover, Jul 25 2017 *)

a(n) = 17^n*cos(2*n*arctan(1/4)).
A recursive formula for T(n) = tan(2*n*arctan(1/4)) is T(n+1) = (8/15+T(n))/(1-8/15*T(n)). Unsigned a(n) is the absolute value of denominator of T(n). [And a(n) = 17^n*cos(n*arctan(8/15)). - Peter Luschny, Sep 29 2019]
From Colin Barker, Jul 25 2017: (Start)
G.f.: x*(15 - 289*x) / (1 - 30*x + 289*x^2).
a(n) = ((15 - 8*i)^n + (15 + 8*i)^n)/2 where i=sqrt(-1).
a(n) = 30*a(n-1) - 289*a(n-2) for n>2. (End)
a(n) = Re((8 + 15*i)^n) = Re((4 + i)^(2*n)) = (1/2)*V(2*n,P = 8,Q = 17), where V(n,P,Q) denotes the Lucas sequence of the second kind and i=sqrt(-1). - Peter Bala, Sep 24 2019

A277557 The ordered image of the 1-to-1 mapping of an integer ordered pair (x,y) into an integer using Cantor's pairing function, where 0 < x < y, gcd(x,y)=1 and x+y odd.

Original entry on oeis.org

8, 18, 19, 32, 33, 34, 50, 52, 53, 72, 73, 74, 75, 76, 98, 99, 100, 101, 102, 103, 128, 131, 133, 134, 162, 163, 164, 165, 166, 167, 168, 169, 200, 201, 202, 203, 204, 205, 206, 207, 208, 242, 244, 247, 248, 250, 251, 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 338

Offset: 1

Frank M Jackson, Oct 19 2016

nonn

The mapping of the ordered pair (x,y) to an integer uses Cantor's pairing function to generate the integer as (x+y)(x+y+1)/2+y. Also for every ordered pair (x,y) such that 0 < x < y, gcd(x,y)=1 and x+y odd, there exists a primitive Pythagorean triple (PPT) (a, b, c) such that a = y^2-x^2, b = 2xy, c = x^2+y^2. Therefore each term in the sequence represents a unique PPT.
Numbers n for which 0 < A025581(n) < A002262(n) and A025581(n)+A002262(n) is odd, and gcd(A025581(n), A002262(n)) = 1. [The definition expressed with A-numbers.] - Antti Karttunen, Nov 02 2016
See also the triangle T(y, x) with the values for PPTs given in A278147. - Wolfdieter Lang, Nov 24 2016

			a(5)=33 because the ordered pair (2,5) maps to 33 by Cantor's pairing function (see below) and is the 5th such occurrence. Also x=2, y=5 generates a PPT with sides (21,20,29).
Note: Cantor's pairing function is simply A001477 in its two-argument tabular form A001477(k, n) = n + (k+n)*(k+n+1)/2, thus A001477(2,5) = 5 + (2+5)*(2+5+1)/2 = 33. - _Antti Karttunen_, Nov 02 2016

Wikipedia, Cantor's pairing function, and Pythagorean triple

Cf. A001477, A002262, A025581, A243808, A277632.
Cf. A020882 (is obtained when A048147(a(n)) is sorted into ascending order), A008846 (same with duplicates removed).
Cf. A020887, A020888, A120427, A024362, A024406, A046079, A046087, A070151, A156678, A156679, A156680, A156683, A156685, A222946, A278147.

Cantor[{i_, j_}] := (i+j)(i+j+1)/2+j; getparts[n_] := Reverse@Select[Reverse[IntegerPartitions[n, {2}], 2], GCD@@#==1 &]; pairs=Flatten[Table[getparts[2n+1], {n, 1, 20}], 1]; Table[Cantor[pairs[[n]]], {n, 1, Length[pairs]}]

A067358 Imaginary part of (5+12i)^n.

Original entry on oeis.org

0, 12, 120, -828, -28560, -145668, 3369960, 58317492, 13651680, -9719139348, -99498527400, 647549275812, 23290743888720, 123471611274972, -2701419604443960, -47880898349909868, -22269070348069440, 7869181117654073292, 82455284065364468280, -505338768229893703548

Offset: 0

Barbara Haas Margolius, (b.margolius(AT)csuohio.edu), Jan 17 2002

Also 13^n sin(2n arctan(2/3)) or numerator of tan(2n arctan(2/3)).

Note that a(n), A067359(n) and 13^n are primitive Pythagorean triples with hypotenuse 13^n.

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 430-433.

J. M. Borwein and R. Girgensohn, Addition theorems and binary expansions, Canadian J. Math. 47 (1995) 262-273.
E. Eckert, The group of primitive Pythagorean triangles, Mathematics Magazine 57 (1984) 22-27.
Steven R. Finch, Plouffe's Constant [Broken link]
Steven R. Finch, Plouffe's Constant [From the Wayback machine]
Simon Plouffe, The Computation of Certain Numbers Using a Ruler and Compass, J. Integer Seqs. Vol. 1 (1998), #98.1.3.
Index entries for linear recurrences with constant coefficients, signature (10,-169).

Cf. A067359 (13^n cos(2n arctan(2/3))).

Cf. A066770, A066771, A067360, A067361, A020888, A014498, A020892.

a[1] := 12/5; for n from 1 to 40 do a[n+1] := (12/5+a[n])/(1-12/5*a[n]):od: seq(abs(numer(a[n])), n=1..40);# a[n]=tan(2n arctan(2/3))

Im[(5 + 12*I)^Range[0, 24]] (* or *)
LinearRecurrence[{10, -169}, {0, 12}, 25] (* Paolo Xausa, Apr 22 2024 *)

PARI
```
a(n)=imag((5+12*I)^n)
```

G.f.: 12*x/(1-10*x+169*x^2). a(n)=10*a(n-1)-169*a(n-2). - Michael Somos, Jun 27 2002

Better description from Michael Somos, Jun 27 2002

A067359 Real part of (5 + 12i)^n.

Original entry on oeis.org

1, 5, -119, -2035, -239, 341525, 3455641, -23161315, -815616479, -4241902555, 95420159401, 1671083125805, 584824319281, -276564805068235, -2864483360640839, 18094618450123325, 665043872449535041, 3592448206424508485, -76467932379726337079, -1371803070683005304755

Offset: 1

Barbara Haas Margolius, (b.margolius(AT)csuohio.edu), Jan 17 2002

Also 13^n*cos(2*n*arctan(2/3)) or denominator of tan(2*n*arctan(2/3)).

Note that A067358(n), a(n) and 13^n are primitive Pythagorean triples with hypotenuse 13^n.

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 430-433.

J. M. Borwein and R. Girgensohn, Addition theorems and binary expansions, Canadian J. Math. 47 (1995) 262-273.
E. Eckert, The group of primitive Pythagorean triangles, Mathematics Magazine 57 (1984) 22-27.
Steven R. Finch, Plouffe's Constant [Broken link]
Steven R. Finch, Plouffe's Constant [From the Wayback machine]
Simon Plouffe, The Computation of Certain Numbers Using a Ruler and Compass, J. Integer Seqs. Vol. 1 (1998), #98.1.3.
Index entries for linear recurrences with constant coefficients, signature (10, -169).

Cf. A067358 (13^n sin(2n arctan(2/3))).

Cf. A066770, A066771, A067360, A067361, A020888, A014498, A020892.

a[1] := 12/5; for n from 1 to 40 do a[n+1] := (12/5+a[n])/(1-12/5*a[n]):od: seq(abs(denom(a[n])), n=1..40);# a[n]=tan(2n arctan(2/3))

Table[Re[(5+12I)^n],{n,0,20}] (* Harvey P. Dale, Aug 24 2014 *)

PARI
```
a(n)=real((5+12*I)^n)
```

From Michael Somos, Jun 27 2002: (Start)
G.f.: (1-5*x)/(1-10*x+169*x^2).
a(n) = 10*a(n-1) - 169*a(n-2). (End)

Better description from Michael Somos, Jun 27 2002

cp's OEIS Frontend

A119321 Hypotenuse of primitive Pythagorean triangles sorted on inradius (A020888), then on hypotenuse.

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A058529 Numbers whose prime factors are all congruent to +1 or -1 modulo 8.

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A014498 Varying radii of inscribed circles within primitive Pythagorean triples as a function of increasing values of hypotenuse.

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A020887 Ordered set of a + b - c as (a,b,c) runs through all primitive Pythagorean triples with a

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A120427 For each y >= 1 there are only finitely many values of x >= 1 such that x-y and x+y are both squares; list all such pairs (x,y) with gcd(x,y) = 1 ordered by values of y; sequence gives y values.

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A067360 a(n) = 17^n sin(2n arctan(1/4)) or numerator of tan(2n arctan(1/4)).

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A067361 a(n) = 17^n*cos(2*n*arctan(1/4)) or denominator of tan(2*n*arctan(1/4)).

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Maple

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A277557 The ordered image of the 1-to-1 mapping of an integer ordered pair (x,y) into an integer using Cantor's pairing function, where 0 < x < y, gcd(x,y)=1 and x+y odd.

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A067361 a(n) = 17^ncos(2n*arctan(1/4)) or denominator of tan(2narctan(1/4)).