A014498 -id:A014498 - cp's OEIS frontend

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

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

A020888 Ordered set of (a + b - c)/2 as (a,b,c) runs through all primitive Pythagorean triples with a

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 5, 6, 6, 7, 7, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 15, 15, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 30, 30, 31, 31, 32, 33, 33, 33, 33, 34, 34, 35, 35, 35, 35

Offset: 1

Clark Kimberling

nonn

n appears A068068(n) number of times. - Lekraj Beedassy, May 03 2006

Ordered inradii of primitive Pythagorean triangles. - Lekraj Beedassy, May 08 2006

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

For values ordered by hypotenuse, see A014498.

a(n) = A020887(n)/2.

Offset corrected to 1 by Ray Chandler, Jan 23 2020

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

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

A087459 Values (X + Y - Z) sorted on Z, then on Y, where (X,Y,Z) is a primitive Pythagorean triple with X

Original entry on oeis.org

2, 4, 6, 6, 12, 10, 8, 20, 10, 24, 14, 30, 28, 12, 30, 40, 18, 42, 14, 36, 56, 22, 16, 42, 60, 70, 44, 18, 72, 48, 70, 26, 84, 66, 90, 20, 52, 80, 88, 30, 78, 22, 60, 90, 110, 112, 60, 126, 104, 24, 66, 132, 34, 126, 130, 144, 68, 26, 154, 120, 110, 140, 156, 38, 102, 28

Offset: 1

Lekraj Beedassy, Oct 23 2003

nonn

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

For ordered values of (X + Y - Z) see A020887.

Cf. A046086, A046087, A020882, A014498.

a(n) = A046086(n) + A046087(n) - A020882(n) = 2*A014498(n).

a(n) = sqrt{2*A118961(n)*A118962(n)}. - Lekraj Beedassy, May 11 2006

Corrected and extended by Ray Chandler, Oct 25 2003

A119322 Inradius of primitive Pythagorean triangles sorted on hypotenuse (A020882), then on inradius.

Original entry on oeis.org

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

Offset: 1

Lekraj Beedassy, May 14 2006

nonn

Ray Chandler, Table of n, a(n) for n = 1..10000
Mohammad K. Azarian, Solution of problem 125: Circumradius and Inradius, Math Horizons, Vol. 16, No. 2 (Nov. 2008), p. 32.

Cf. A119321, A014498 (different order).

Extended by Ray Chandler, Apr 12 2010

A056901 Least semiperimeter s of primitive Pythagorean triangle with inradius n.

Original entry on oeis.org

6, 15, 20, 45, 42, 35, 72, 153, 110, 63, 156, 77, 210, 99, 88, 561, 342, 143, 420, 117, 130, 195, 600, 209, 702, 255, 812, 165, 930, 187, 1056, 2145, 238, 399, 204, 221, 1482, 483, 304, 273, 1806, 247, 1980, 285, 266, 675, 2352, 665, 2550, 783, 460, 357

Offset: 1

Lekraj Beedassy, Feb 12 2002

nonn

For a primitive Pythagorean triangle with sides X, Y & Z, we have two generating numbers m&n such that m>n, gcd(m,n) = 1 and the parity of m&n are opposite. X = m^2 - n^2, Y = 2mn and Z = m^2 + n^2, s = m^2 + mn and finally r = n(m-n).

Moreover, a primitive Pythagorean triangle has area n*a(n).

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.
Albert H. Beiler, "Recreations In The Theory Of Numbers, The Queen Of Mathematics Entertains," Dover Publications, Inc., Second Edition, NY, 1966, Chapter XIV, 'The Eternal Triangle,' pages 104 - 134.

Eric Weisstein's World of Mathematics, Semiperimeter
Wm. H. Richardson, The inradius of a Right Triangle with Integral Sides

Cf. A014498.

a = Table[10^9, {75} ]; Do[ If[ GCD[m, n] == 1 && Sort[ Mod[ {m, n}, 2]] == {0, 1}, s = m^2 + m*n; r = n(m - n); If[r < 76 && a[[r]] > s, a[[r]] = s; Print[r, " ", s]]], {m, 2, 10^2}, {n, 1, m - 1} ]

When n is (i) an odd prime power, s = (n + 1)(n + 2). (ii) a power of 2, s = (n + 1)(2n + 1). (iii) a composite with relatively prime factors a*b such that a is smallest, s = (a + b)(2a + b).

Edited and extended by Robert G. Wilson v, Feb 18 2002

cp's OEIS Frontend

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

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A020888 Ordered set of (a + b - c)/2 as (a,b,c) runs through all primitive Pythagorean triples with a

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

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

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Maple

Mathematica

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A087459 Values (X + Y - Z) sorted on Z, then on Y, where (X,Y,Z) is a primitive Pythagorean triple with X

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