A229083 -id:A229083 - cp's OEIS frontend

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

Ralf Stephan, Sep 14 2013

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.

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

Cf. A229117, A000217, A000290.

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 *)

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, ","))

A229131 Numbers k such that the distance between the k-th triangular number and the nearest square is exactly 1.

Original entry on oeis.org

1, 2, 4, 5, 15, 25, 32, 90, 148, 189, 527, 865, 1104, 3074, 5044, 6437, 17919, 29401, 37520, 104442, 171364, 218685, 608735, 998785, 1274592, 3547970, 5821348, 7428869, 20679087, 33929305, 43298624

Offset: 1

Ralf Stephan, Sep 15 2013

nonn

The k-th triangular number (A000217(k)) is a square plus or minus one.

Union of A006451 (k-th triangular number is a square minus one) and A072221 (k-th triangular number is a square plus one).

			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.

Cf. A000217, A001110, A006451, A072221, A229083, A229118.

for(n=1,10^8,for(i=-1,1,f=0;if(i&&issquare(n*(n+1)/2+i),f=1;break));if(f,print1(n,",")))

G.f.: (-x^7 + 2*x^6 - 2*x^5 + 4*x^4 - 5*x^3 + 2*x^2 + x + 1)/((1-6*x^3+x^6)*(1-x)) (conjectured).

cp's OEIS Frontend

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

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