A229118 Distance from the n-th triangular number to the nearest square.

0, 1, 2, 1, 1, 4, 3, 0, 4, 6, 2, 3, 9, 5, 1, 8, 9, 2, 6, 14, 6, 3, 13, 11, 1, 10, 17, 6, 6, 19, 12, 1, 15, 19, 5, 10, 26, 12, 4, 21, 20, 3, 15, 29, 11, 8, 28, 20, 0, 21, 30, 9, 13, 36, 19, 4, 28, 30, 6, 19, 42, 17, 9, 36, 29, 2, 26, 42, 14, 15, 45, 27, 3, 34, 41, 10, 22, 55, 24, 9, 43, 39, 5, 30, 55, 20, 16, 53, 36, 1

Offset: 1

Author

Ralf Stephan, Sep 14 2013

Keywords

• nonn
• easy

Comments

The maximum of a(n)/n appears to converge to sqrt(2)/2 (A010503), i.e. n*(n+1)/2 seems not more than n*sqrt(2)/2 distant from a square.

Some values don't seem to be in the sequence (checked up to n=10^7): 7,18,23,31,37,38...

Those values k are not in the sequence because the Pell-type equations x^2 - 8*y^2 = 8*k+1 and x^2 - 8*y^2 = -8*k+1 have no solutions. - Robert Israel, Apr 08 2019

a(A001108(n)) = 0, a(A229131(n)) = 1, a(A229083(n)) <= 1, a(A229133(n)) is square.

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A229117, A000217, A000290.

Programs

Mathematica

dns[n_]:=Module[{a=Floor[Sqrt[n]]^2,b=Ceiling[Sqrt[n]]^2},Min[n-a, b-n]]; dns/@Accumulate[Range[90]] (* Harvey P. Dale, Nov 07 2016 *)

PARI

m=0;for(n=1, 100, t=n*(n+1)/2;s=sqrtint(t);d=min(t-s^2,(s+1)^2-t);print1(d, ","))