A229083 Numbers k such that the distance between the k-th triangular number and the nearest square is at most 1.

1, 2, 4, 5, 8, 15, 25, 32, 49, 90, 148, 189, 288, 527, 865, 1104, 1681, 3074, 5044, 6437, 9800, 17919, 29401, 37520, 57121, 104442, 171364, 218685, 332928, 608735, 998785, 1274592, 1940449, 3547970, 5821348, 7428869, 11309768, 20679087, 33929305, 43298624, 65918161

Offset: 1

Author

Ralf Stephan, Sep 13 2013

Keywords

• nonn

Comments

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

Union of A006451 (k-th triangular number is a square minus one), A072221 (k-th triangular number is a square plus one), and A001108 (k-th triangular number is square). Also, union of A229131 and A001108.

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.
A000217(8) = 36 = 6^2 so 8 is in sequence.
		

Crossrefs

Cf. A000217, A001108, A006451, A072221, A229118, A229131.

Programs

PARI

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

Formula

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