A229117 Numbers k where d/k reaches a new record, with d the distance from the k-th triangular number to the nearest square.

2, 3, 13, 20, 37, 78, 119, 218, 457, 696, 1273, 2666, 4059, 7422, 15541, 23660, 43261, 90582, 137903, 252146, 527953, 803760, 1469617, 3077138, 4684659, 8565558, 17934877, 27304196, 49923733, 104532126, 159140519, 290976842

Offset: 1

Author

Ralf Stephan, Sep 14 2013

Keywords

• nonn

Comments

Positions of records of A229118(n)/n.

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

Examples

			G.f. = 2*x + 3*x^2 + 13*x^3 + 20*x^4 + 37*x^5 + 78*x^6 + 119*x^7 + 218*x^8 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..2500

Crossrefs

Cf. A010503, A229118.

Programs

Magma

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(2 +x+10*x^2-5*x^3+11*x^4-19*x^5+x^6-2*x^7+3*x^8)/(1-x-6*x^3+6*x^4+x^6- x^7))); // G. C. Greubel, Aug 09 2018

Mathematica

Drop[CoefficientList[Series[x*(2 + x + 10*x^2 - 5*x^3 + 11*x^4 - 19*x^5 + x^6 - 2*x^7 + 3*x^8)/(1 - x - 6*x^3 + 6*x^4 + x^6 - x^7), {x, 0, 50}], x], 1] (* G. C. Greubel, Aug 09 2018 *)

PARI

m=0;for(n=1, 10^9, t=n*(n+1)/2;s=sqrtint(t);d=min(t-s^2,(s+1)^2-t);r=d/n;if(r>m,m=r;print1(n, ",")))

PARI

{a(n) = if( n<1, 0, polcoeff( (1 + x + x^2 + 4*x^3 + x^4 + 11*x^5 - 18*x^6 - 2*x^8 + 3*x^9) / (1 - x - 6*x^3 + 6*x^4 + x^6 - x^7) + x * O(x^n), n))}; /* Michael Somos, Dec 25 2016 */

Formula

G.f.: x * (2 + x + 10*x^2 - 5*x^3 + 11*x^4 - 19*x^5 + x^6 - 2*x^7 + 3*x^8) / (1 - x - 6*x^3 + 6*x^4 + x^6 - x^7). - Michael Somos, Dec 25 2016

a(n) = a(n-1) + 6*a(n-3) - 6*a(n-4) - a(n-6) + a(n-7) if n>9. - Michael Somos, Dec 25 2016