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 A229118 Distance from the n-th 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified October 17 04:09 EDT 2019. Contains 328106 sequences. (Running on oeis4.)