A331261 List of pairs of numbers having certain properties (see Comments).

0, 1, 4, 5, 24, 25, 144, 145, 840, 841, 4900, 4901, 28560, 28561, 166464, 166465, 970224, 970225, 5654884, 5654885, 32959080, 32959081, 192099600, 192099601, 1119638520, 1119638521, 6525731524, 6525731525, 38034750624, 38034750625, 221682772224, 221682772225

Offset

1,3

Comments

List of pairs of numbers with following properties:

(1) The numbers in each pair are consecutive.

(2) At least one integer is a square.

(3) The sum of each pair is another square.

(4) The position of the square term oscillates between left and right.

(5) The triple formed by the pair and the square root of their sum is a Pythagorean triple.

Links

Table of n, a(n) for n=1..32.

Prog

(Python)
# a Python 3.7 program to arrive at Pell-Fermat type Pythagorean triples below a given/entered number.
import math
number = None
# Taking the input from user
number = 10000000
squareList = []
subPythagoreanTriple= []
needed = 1.0
while(needed <= number):
    root = math.sqrt(needed)
    if int(root + 0.5) ** 2 == needed:
        squareList.append(needed)
        flag = True
    else:
        #print(number, "is not a perfect square")
        pass
        flag = None
    needed = needed + 1
def checkSubPythagoreanTriple():
    for i in squareList:
        SPTless = i-1
        SPTmore = i+1
        firstSide = 2*i - 1
        secondSide = 2*i + 1
        firstSideRoot = math.sqrt(firstSide)
        secondSideRoot = math.sqrt(secondSide)
        if int(firstSideRoot + 0.5) ** 2 == firstSide or int(secondSideRoot + 0.5) ** 2 == secondSide :
            if(i%2 == 1):
                subPythagoreanTriple.append(int(i-1))
                subPythagoreanTriple.append(int(i))
            else:
                subPythagoreanTriple.append(int(i))
                subPythagoreanTriple.append(int(i+1))
        else:
            pass
def main():
    checkSubPythagoreanTriple()
main()
print(str(subPythagoreanTriple))
(Magma) a:=[0, 1]; for k in [2..1000000] do if IsSquare(2*k*k+1) then a:=a cat [k^2, k^2+1]; else if IsSquare(2*k*k-1) then a:=a cat [k^2-1, k^2]; end if; end if; end for; a; // Marius A. Burtea, Jan 21 2020

Crossrefs

Sequence in context: A361981 A338422 A171885 * A063986 A039583 A042123

Adjacent sequences: A331258 A331259 A331260 * A331262 A331263 A331264

Keyword

nonn

Author

Chandrasekhar Karri, Jan 13 2020

Extensions

More terms from Marius A. Burtea, Jan 13 2020

Status

approved