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

1,2

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.

Links

Table of n, a(n) for n=1..41.

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).

Example

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.

Prog

(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, ", ")))

Crossrefs

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

Sequence in context: A184775 A240075 A272229 * A194415 A364779 A293536

Adjacent sequences: A229080 A229081 A229082 * A229084 A229085 A229086

Keyword

nonn

Author

Ralf Stephan, Sep 13 2013

Status

approved