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A229117 Numbers n where d/n reaches a new record, with d the distance of the n-th triangular number to the nearest square. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Positions of records of A229118(n)/n.

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

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

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Last modified March 29 18:37 EDT 2020. Contains 333117 sequences. (Running on oeis4.)