A267077 Least m>0 for which m*n^2 + 1 is a square and m*triangular(n) + 1 is a triangular number (A000217). Or -1 if no such m exists.

1, 35, 30, 18135, 189, 27, 321300, 23760, 1188585957, 1656083, 26, 244894427400, 82093908624206325, 1858717755529547, 86478, 21491811639746039592, 26135932603458945934958445, 353382195058506640426335, 26780050, 7859354769338288038121982384, 274554988002

Offset: 0

Author

Alex Ratushnyak, Jan 10 2016

Keywords

• nonn

Examples

			26*10^2+1 = 2601 is a square, and 26*10*11/2+1 = 1431 = triangular(53), and 26 is the smallest such multiplier, therefore a(10) = 26.
		

Links

• Don Reble, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A000217, A000290, A035096, A061782, A067872, A188621.

Programs

Python

from math import sqrt
def A267077(n):
    if n == 0:
        return 1
    u,v,t,w = max(8,2*n),max(4,n)**2-9,4*n*(n+1),n**2
    while True:
        m,r = divmod(v,t)
        if not r and int(sqrt(m*w+1))**2 == m*w+1:
            return m
        v += u+1
        u += 2 # Chai Wah Wu, Jan 15 2016

Python

#!/usr/bin/python3
# This sequence is easy if you use a Pell-equation solver such as labmath.py
# Solve the A267077 Pell equation:
# nx^2 - (4n+4)y^2 = 5n-4; but also y^2 == 1 mod n^2
# Let u = nx, then # u^2 - n*(4n+4)y^2 = n*(5n-4)
#   and (y > n) and (u == 0 mod n) and (y^2 == 1 mod n^2)
# (y > n makes m > 0)
# Report m = (y^2 - 1) / n^2
import labmath
print(0, 1)
print(1, 35) # When n<2, the Pell equation is elliptical.
for nn in range(2,1001):
    nsq = nn * nn
    ps = labmath.pell(nn*(4*nn+4), nn*(5*nn-4))
    uu,yy = next(ps[0])
    while (yy <= nn) or ((uu % nn) != 0) or ((yy*yy) % nsq != 1):
        uu,yy = next(ps[0])
    print(nn, (yy*yy - 1) // nsq)
# From Don Reble, Apr 15 2022, added by N. J. A. Sloane, Apr 15 2022.

Extensions

a(12)-a(15) from Chai Wah Wu, Jan 16 2016

a(16) and beyond from Don Reble, Apr 15 2022