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

1,3

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

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 *)

Prog

(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, ", "))

Crossrefs

Cf. A229117, A000217, A000290.

Sequence in context: A112096 A217874 A323182 * A320796 A026725 A026758

Adjacent sequences: A229115 A229116 A229117 * A229119 A229120 A229121

Keyword

nonn,easy

Author

Ralf Stephan, Sep 14 2013

Status

approved