A385972
The long legs of the triangles defined in A365577.
Original entry on oeis.org
4, 24, 480, 130560, 8589803520, 36893488138829168640, 680564733841876926889855726716117319680, 231584178474632390847141970017375815705859404597439251151988418800962722856960
Offset: 1
A385973
The hypotenuses of the triangles defined in A365577.
Original entry on oeis.org
5, 25, 481, 130561, 8589803521, 36893488138829168641, 680564733841876926889855726716117319681, 231584178474632390847141970017375815705859404597439251151988418800962722856961
Offset: 1
A377725
Length of the short leg of the unique primitive Pythagorean triple whose inradius is A002315(n) and such that its long leg and its hypotenuse are consecutive natural numbers.
Original entry on oeis.org
3, 15, 83, 479, 2787, 16239, 94643, 551615, 3215043, 18738639, 109216787, 636562079, 3710155683, 21624372015, 126036076403, 734592086399, 4281516441987, 24954506565519, 145445522951123, 847718631141215, 4940866263896163, 28797478952235759, 167844007449518387
Offset: 1
Triangles begin:
n=1: 3, 4, 5;
n=2: 15, 112, 113;
n=3: 83, 3444, 3445;
n=4: 479, 114720, 114721;
...
This sequence gives the first column.
A053630
Pythagorean spiral: a(n-1), a(n)-1 and a(n) are sides of a right triangle.
Original entry on oeis.org
3, 5, 13, 85, 3613, 6526885, 21300113901613, 226847426110843688722000885, 25729877366557343481074291996721923093306518970391613
Offset: 1
a(3)=13 because 5,12,13 is a Pythagorean triple and a(2)=5.
- R. Gelca and T. Andreescu, Putnam and Beyond, Springer 2007, p. 121.
See also
A018928,
A180313 and
A239381 for similar sequences with a(n) a leg and a(n+1) the hypotenuse of a Pythagorean triangle.
-
A:= proc(n) option remember; (procname(n-1)^2+1)/2 end proc: A(1):= 3:
seq(A(n),n=1..10); # Robert Israel, Jul 14 2014
-
NestList[(#^2+1)/2&,3,10] (* Harvey P. Dale, Sep 15 2011 *)
-
{a(n) = if( n>1, (a(n-1)^2 + 1) / 2, 3)}; /* Michael Somos, May 15 2011 */
A378395
Sequence of primitive Pythagorean triples beginning with the triple (3,4,5), with each subsequent triple having as its inradius the hypotenuse of the previous triple, and with the long leg and the hypotenuse of each triple being consecutive natural numbers.
Original entry on oeis.org
3, 4, 5, 11, 60, 61, 123, 7564, 7565, 15131, 114473580, 114473581, 228947163, 26208401722874284, 26208401722874285, 52416803445748571, 1373760641735119632984407274271020, 1373760641735119632984407274271021
Offset: 1
Triples begin:
3, 4, 5;
11, 60, 61;
123, 7564, 7565;
15131, 114473580, 11447358;
...
- J. M. Blanco Casado, J. M. Sánchez Muñoz, and M. A. Pérez García-Ortega, El Libro de las Ternas Pitagóricas, Preprint 2024.
-
{a0,b0,c0}={3,4,5};f[n_]:=Module[{fn0=2c0+1,fn1=((2c0+1)^2+1)/2},Do[{fn0,fn1}={2fn1+1,((2fn1+1)^2+1)/2},{n}];fn0];t[n_]:={f[n-1],(f[n-1]^2-1)/2,(f[n-1]^2+1)/2};ternas={a0,b0,c0};For[i=1,i<=6,i++,ternas=Join[ternas,t[i]]];ternas
A379506
Sequence of primitive Pythagorean triples beginning with the triple (3,4,5), with each subsequent triple having as its inradius the semiperimeter of the previous triple, and with the long leg and the hypotenuse of each triple being consecutive natural numbers.
Original entry on oeis.org
3, 4, 5, 13, 84, 85, 183, 16744, 16745, 33673, 566935464, 566935465, 1133904603, 642869824352293804, 642869824352293805, 1285739649838492213, 826563223583404284483387832630818684, 826563223583404284483387832630818685
Offset: 1
Triples begin:
3, 4, 5;
13, 84, 85;
183, 16744, 16745;
33673, 566935464, 566935465;
-
{a0,b0,c0}={3,4,5};f[n_]:= Module[{fn0=a0+b0+c0+1,fn1=((a0+b0+c0+1)^2+1)/2},Do[{fn0,fn1}={2fn1+fn0,((2fn1+fn0)^2+1)/2},{n}];fn0];t[n_]:={f[n-1],(f[n-1]^2-1)/2,(f[n 1]^2+1)/2};ternas={a0,b0,c0};For[i=1,i<=6,i++,ternas=Join[ternas,t[i]]];ternas
A378963
Sequence of primitive Pythagorean triples beginning with the triple (3,4,5), with each subsequent triple having as its inradius the short leg of the previous triple, and with the long leg and the hypotenuse of each triple being consecutive natural numbers.
Original entry on oeis.org
3, 4, 5, 7, 24, 25, 15, 112, 113, 31, 480, 481, 63, 1984, 1985, 127, 8064, 8065, 255, 32512, 32513, 511, 130560, 130561, 1023, 523264, 523265, 2047, 2095104, 2095105, 4095, 8384512, 8384513, 8191, 33546240, 33546241, 16383, 134201344, 134201345
Offset: 1
Triples begin:
3, 4, 5;
7, 24, 25;
15, 112, 113;
31, 480, 481;
- Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz, and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2024.
-
a=Table[2^(n+1)-1,{n,1,13}];Apply[Join,Map[{#,(#^2-1)/2,(#^2+1)/2}&,a]]
A380299
Sequence of primitive Pythagorean triples beginning with the triple (3,4,5), with each subsequent triple having as its inradius the area of the previous triple, and with the long leg and the hypotenuse of each triple being consecutive natural numbers.
Original entry on oeis.org
3, 4, 5, 13, 84, 85, 1093, 597324, 597325, 652875133, 213122969644883844, 213122969644883845, 139142687152258502421051253, 9680343693975641657052402486887446135645084826435004, 9680343693975641657052402486887446135645084826435005
Offset: 1
Triples begin:
3, 4, 5;
13, 84, 85;
1093, 597324, 597325;
652875133, 213122969644883844, 213122969644883845;
- El Libro de las Ternas Pitagóricas, Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz y José Miguel Blanco Casado, Preprint, 2025.
-
{a0,b0,c0}={3,4,5};f[n_]:=Module[{fn0=a0 b0+1,fn1=((a0 b0+1)^2-1)/2},Do[{fn0,fn1}={fn1 fn0+1,((fn1 fn0+1)^2-1)/2},{n}];fn0];t[n_]:= {f[n-1],(f[n-1]^2-1)/2,(f[n-1]^2+1)/2};ternas={a0,b0,c0};For[i=1,i<=5,i++,ternas=Join[ternas,t[i]]];ternas
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