A018928 -id:A018928 - cp's OEIS frontend

A018930 a(1)=3; for n>1, a(n) is smallest positive integer such that a(1)^2+...+a(n)^2 = m^2 for some m.

3, 4, 12, 84, 132, 12324, 1836, 105552, 255084, 197580, 10358340, 13775220, 1936434780, 51299286012, 123205977516, 862441842612, 1310543298204, 667510076211780, 207181940072172, 110912831751840, 1698410314006284

Offset: 1

Charles Reed (charles.reed(AT)bbs.ewgateway.org)

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..100

Cf. A018928, A018929, A127689.

f[n_]:=Module[{a={3}}, Do[AppendTo[a,First[y/. {ToRules[Reduce[{y^2+a.a == x^2,x>0,y>0}, {y,x},Integers]]}]], {n-1}]; a]; f[21]//Timing (* Jean-François Alcover, Jan 26 2007 *)

print1("3, "); s=9; for(n=1,30, d=divisors(s); t=d[#d\2]; q=(s\t-t)/2; print1(q,", "); s+=q^2); \\ Max Alekseyev, Nov 23 2012

More terms from David W. Wilson.

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

Henry Bottomley, Mar 21 2000

Least prime factors of a(n): 3, 5, 13, 5, 3613, 5, 233, 5, 3169, 5, 101, 5, 29, 5, 695838629, 5, 1217, 5, 2557, 5, 101, 5, 769, 5. - Zak Seidov, Nov 11 2013

			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.

Steven Finch, Exercises in Iterational Asymptotics, arXiv:2411.16062 [math.NT], 2024. See p. 10.
Miguel-Ángel Pérez García-Ortega, Capitulo 5. Catetos, El Libro de las Ternas Pitagóricas.

Cf. A000058, A001844, A006892.
See also A018928, A180313 and A239381 for similar sequences with a(n) a leg and a(n+1) the hypotenuse of a Pythagorean triangle.
Cf. A077125, A117191 (4^(1/Pi)).

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 */

a(1) = 3, a(n) = (a(n-1)^2 + 1)/2 for n > 1.
a(n) = 2*A000058(n)-1 = A053631(n)+1 = floor(2 * 1.597910218031873...^(2^n)). Constructing the spiral as a sequence of triangles with one vertex at the origin, then for large n the other vertices are close to lying on the doubly logarithmic spiral r = 2*2.228918357655...^(1.5546822754821...^theta) where theta(n) = n*Pi/2 - 1.215918200344... and 1.5546822754821... = 4^(1/Pi).
a(1) = 3, a(n+1) = (1/4)*((a(n)-1)^2 + (a(n)+1)^2). - Amarnath Murthy, Aug 17 2005
a(n)^2 - (a(n)-1)^2 = a(n-1)^2, so 2*a(n)-1 = a(n-1)^2 (see the first formula). - Thomas Ordowski, Jul 13 2014
a(n) = (A006892(n+2) + 3)/2. - Thomas Ordowski, Jul 14 2014
a(n)^2 = A006892(n+3) + 2. - Thomas Ordowski, Jul 19 2014

Corrected and extended by James Sellers, Mar 22 2000

A018929 Define {b(n)} by b(1) = 3, b(n) (n >= 2) is smallest number such that b(1)^2 + ... + b(n)^2 = m^2 for some m and all b(i) are distinct. Sequence gives values of m^2.

Original entry on oeis.org

9, 25, 169, 7225, 24649, 151905625, 155276521, 11296501225, 76364348281, 115402204681, 107410609760281, 297167295808681, 3750076824489457081, 2635366822165468321225, 17815079717838565851481, 761621011605820344834025

Offset: 1

Charles Reed (charles.reed(AT)bbs.ewgateway.org)

nonn

Cf. A018928, A018930.

a(n) = A018928(n+1)^2 - A018930(n+1)^2. - César Aguilera, Nov 02 2018

More terms from David W. Wilson

A239381 a(0) = 3, the least length of a Primitive Pythagorean Triangle (PPT). a(n) is the least hypotenuse of a PPT which has a(n-1) as one of its legs.

Original entry on oeis.org

3, 5, 13, 85, 157, 12325, 90733, 2449525, 28455997, 295742792965, 171480834409967437, 656310093705697045, 1616599508725767821225590944157, 4461691012090851100342993272805, 115366949386695884000892071602798585632943213, 12002377162350258332845595301471273220420939451301220405

Offset: 0

Robert G. Wilson v, Mar 17 2014

nonn

a(0)=3 because A042965(3)=3 with comments.

If we relax the Primitive restriction, i.e., GCD(x,y,z) can exceed 1, then we have A018928.

			a(0)=3 by definition,
a(1)=5 because it is the hypotenuse of a 3-4-5 PPT,
a(2)=13 because it is the hypotenuse of a 5-12-13 PPT,
a(3)=85 because it is the hypotenuse of a 13-84-85 PPT,
a(4)=157 because it is the hypotenuse of a 85-132-157 PPT, 85 is also the leg of a 85-3612-3613 PPT but its hypotenuse is larger, etc.

Robert G. Wilson v, Table of n, a(n) for n = 0..22

Cf. A008846, A235598.

f[s_List] := Block[{x = s[[-1]]}, Append[s, Transpose[ Solve[x^2 + y^2 == z^2 && GCD[x, y, z] == 1 && y > 0 && z > 0, {y, z}, Integers]][[-1, 1, 2]]]]; lst = Nest[f, {3}, 15]

A180313 A sequence a(n) such that a(n+1)^2 - a(n)^2 are perfect squares.

Original entry on oeis.org

3, 5, 13, 85, 221, 1445, 3757, 24565, 63869, 417605, 1085773, 7099285, 18458141, 120687845, 313788397, 2051693365, 5334402749, 34878787205, 90684846733, 592939382485, 1541642394461, 10079969502245, 26207920705837, 171359481538165, 445534651999229, 2913111186148805

Offset: 1

Valentin Tiriac (valtron2000(AT)gmail.com), Aug 26 2010

nonn

The lexically smallest sequence with a(n+1)^2-a(n)^2 representing perfect squares is A018928.

This version here is constructed via a(n+1) = a(n)* sqrt( 1+((p^2-1)/(2p))^2) where p = A020639(a(n)) is the smallest prime divisor of the previous term.

			After a(1)=3, p=3 (again) and a(2) = 3*sqrt(1+ (8/6)^2) = 5.
After a(4)=85, p=5 and a(5) = 85*sqrt(1+ (24/10)^2) = 85*sqrt(169/25) = 221.

A020639 := proc(n) min(op(numtheory[factorset](n))) ; end proc:
A180313 := proc(n) option remember; if n = 1 then 3; else aprev := procname(n-1) ; p := A020639(aprev) ; aprev* sqrt(1+((p^2-1)/2/p)^2) ; end if; end proc:
for n from 1 to 30 do printf("%d,",A180313(n)) ; end do: # R. J. Mathar, Sep 23 2010

spd[n_] := FactorInteger[n][[1, 1]];
a[n_] := a[n] = If[n == 1, 3, aprev = a[n-1];
    p = spd[aprev]; aprev*Sqrt[1+((p^2-1)/2/p)^2]];
Table[a[n], {n, 1, 26}] (* Jean-François Alcover, Feb 28 2024, after R. J. Mathar *)

# use 5.12.0; use warnings; use Math::Prime::TiedArray; tie my @primes, 'Math::Prime::TiedArray';
sub SmallestPrimeDivisor ($) { my ($n) = @_; for my $p (@primes) { if ($n % $p == 0) { return $p; } } }
sub FindIncrement ($) { my ($n) = @_; my $p = SmallestPrimeDivisor $n; my $k = $n / $p; return $k * ($p ** 2 - 1) / 2; }
my $n = 3; say $n; for my $i (0 .. 23) { my $d = FindIncrement $n; $n = sqrt($d ** 2 + $n ** 2); say $n; }

Nomenclature normalized by R. J. Mathar, Sep 23 2010

Corrected indexing error introduced with previous edit - R. J. Mathar, Oct 01 2010

cp's OEIS Frontend

A018930 a(1)=3; for n>1, a(n) is smallest positive integer such that a(1)^2+...+a(n)^2 = m^2 for some m.

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A053630 Pythagorean spiral: a(n-1), a(n)-1 and a(n) are sides of a right triangle.

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A018929 Define {b(n)} by b(1) = 3, b(n) (n >= 2) is smallest number such that b(1)^2 + ... + b(n)^2 = m^2 for some m and all b(i) are distinct. Sequence gives values of m^2.

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A239381 a(0) = 3, the least length of a Primitive Pythagorean Triangle (PPT). a(n) is the least hypotenuse of a PPT which has a(n-1) as one of its legs.

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A180313 A sequence a(n) such that a(n+1)^2 - a(n)^2 are perfect squares.

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