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

1,1

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.

Links

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

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

Example

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 + ...

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 *)

Prog

(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 */
(Magma) m:=50; R<x>:=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

Crossrefs

Cf. A010503, A229118.

Sequence in context: A084958 A249573 A295111 * A295722 A037392 A136341

Adjacent sequences: A229114 A229115 A229116 * A229118 A229119 A229120

Keyword

nonn

Author

Ralf Stephan, Sep 14 2013

Status

approved