

A229118


Distance from the nth triangular number to the nearest square.


5



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
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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 Pelltype 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[na, bn]]; 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(ts^2, (s+1)^2t); 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



