A076604 -id:A076604 - cp's OEIS frontend

A077036 Duplicate of A076604.

11, 60, 61, 1860, 1861, 1731660, 1731661, 1499324909460, 1499324909461

Offset: 1

dead

A077034 a(1)=3; a(2n), a(2n+1) are smallest integers > a(2n-1) such that a(2n-1)^2+a(2n)^2=a(2n+1)^2.

3, 4, 5, 12, 13, 84, 85, 132, 157, 12324, 12325, 15960, 20165, 26280, 33125, 79500, 86125, 95400, 128525, 152040, 199085, 477804, 517621, 871500, 1013629, 513721874820, 513721874821, 4351526469072, 4381745402885, 10516188966924, 11392538047501

Offset: 1

Zak Seidov, Oct 21 2002

nonn

Note that each time two more terms are added simultaneously. The sequence is infinite.
Smallest sequence of Pythagorean triples {a(k-1),a(k),a(k+1)},with k=2n,such that the hypotenuse of one triangle is the short leg of the next one. Such a sequence is called 2-prime Pythagorean because only the first two triangles (3,4,5),(5,12,13) both have prime hypotenuse and short leg. The next such sequence is given by A076604. Actually, the starting terms for all 2-prime and 3-prime Pythagorean triangles are given respectively by A048270 and A048295. The starting term for the smallest n-prime Pythagorean triangle is A105318. - Lekraj Beedassy, Sep 16 2005
a(2n) <= (a(2n-1)^2-1)/2; a(2n+1) <= (a(2n-1)^2+1)/2. [Max Alekseyev, May 11 2011]

			a(1)=3 implies a(2)=4 and a(3)=5: 3^2+4^2=5^2.
a(3)=5 implies a(4)=12 and a(5)=13: 5^2+12^2=13^2.

Cf. A048270, A048295, A076604, A105318.

More terms from Max Alekseyev, May 11 2011

A304207 a(1)=17; for n>1, a(n) = (a(n-1)^2 - 1)/2 if n is even, a(n-1) + 1 if n is odd.

Original entry on oeis.org

17, 144, 145, 10512, 10513, 55261584, 55261585, 1526921388356112, 1526921388356113, 1165744463109679828308252234384, 1165744463109679828308252234385, 679480076635437837059150531810555804350472205781672488164112, 679480076635437837059150531810555804350472205781672488164113

Offset: 1

Benjamin Knight, May 08 2018

nonn

{17, 144, 145}, {145, 10512, 10513}, {10513, 55261584, 55261585}, ... are sides {a < b < c} of the right triangles, with hypotenuse c = b + 1.

Eric Chen, Table of n, a(n) for n = 1..21

Same basic form as A076601, A076602, A076603, and A076604.

nxt[{n_,a_}]:={n+1,If[OddQ[n],(a^2-1)/2,a+1]}; NestList[nxt,{1,17},20][[All,2]] (* Harvey P. Dale, Mar 27 2021 *)

a(n) = if(n==1,17,if(n%2,a(n-1)+1,(a(n-1)^2 - 1)/2)) \\ Eric Chen, Jun 09 2018

A077035 a(1)=7; a(n),a(n+1) are smallest > a(n-1) such that a(n-1)^2+a(n)^2=a(n+1)^2.

Original entry on oeis.org

7, 24, 25, 60, 65, 72, 97, 4704, 4705, 11292, 12233, 79044, 79985, 124212, 147737, 430416, 455065, 504072, 679097, 24502296, 24511705, 34278300, 42140545, 68012700, 80009705, 192023292, 208025233, 356427144, 412692145, 990461148, 1072999577, 2403086064, 2631758105

Offset: 1

Zak Seidov, Oct 21 2002

nonn

Note that each time two more terms are added simultaneously.

			a(1)=7 therefore a(2)=24 and a(3)=25: 7^2+24^2=25^2; a(3)=25 therefore a(4)=60 and a(5)=65: 25^2+60^2=65^2.

Cf. A077034, A076604.

from math import isqrt
from sympy.ntheory.primetest import is_square
def aupton(terms):
    alst = [7]
    for n in range(2, terms+1, 2):
        sq1, an = alst[-1]**2, alst[-1] + 1
        while not is_square(sq1 + an**2): an += 1
        alst.extend([an, isqrt(sq1 + an**2)])
    return alst[:terms]
print(aupton(19)) # Michael S. Branicky, Jul 24 2021

a(16) and beyond from Michael S. Branicky, Jul 24 2021

cp's OEIS Frontend

A077036 Duplicate of A076604.

Views

Author

Keywords

A077034 a(1)=3; a(2n), a(2n+1) are smallest integers > a(2n-1) such that a(2n-1)^2+a(2n)^2=a(2n+1)^2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

A304207 a(1)=17; for n>1, a(n) = (a(n-1)^2 - 1)/2 if n is even, a(n-1) + 1 if n is odd.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A077035 a(1)=7; a(n),a(n+1) are smallest > a(n-1) such that a(n-1)^2+a(n)^2=a(n+1)^2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Python

Extensions