A229131 Numbers k such that the distance between the k-th triangular number and the nearest square is exactly 1.

1, 2, 4, 5, 15, 25, 32, 90, 148, 189, 527, 865, 1104, 3074, 5044, 6437, 17919, 29401, 37520, 104442, 171364, 218685, 608735, 998785, 1274592, 3547970, 5821348, 7428869, 20679087, 33929305, 43298624

Offset: 1

Author

Ralf Stephan, Sep 15 2013

Keywords

• nonn

Comments

The k-th triangular number (A000217(k)) is a square plus or minus one.

Union of A006451 (k-th triangular number is a square minus one) and A072221 (k-th triangular number is a square plus one).

Examples

			A000217(4)=10 and 10 - 3^2 = 1 so 4 is in the sequence.
A000217(5)=15 and 4^2 - 15 = 1 so 5 is in the sequence.
		

Crossrefs

Cf. A000217, A001110, A006451, A072221, A229083, A229118.

Programs

PARI

for(n=1,10^8,for(i=-1,1,f=0;if(i&&issquare(n*(n+1)/2+i),f=1;break));if(f,print1(n,",")))

Formula

G.f.: (-x^7 + 2*x^6 - 2*x^5 + 4*x^4 - 5*x^3 + 2*x^2 + x + 1)/((1-6*x^3+x^6)*(1-x)) (conjectured).