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 A154154 Indices n such that 30 plus the n-th triangular number is a perfect square. 2
 3, 13, 34, 84, 203, 493, 1186, 2876, 6915, 16765, 40306, 97716, 234923, 569533, 1369234, 3319484, 7980483, 19347373, 46513666 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 LINKS F. T. Adams-Watters, SeqFan Discussion, Oct 2009 FORMULA {k: 30+k*(k+1)/2 in A000290}. Conjectures: (Start) a(n) = +a(n-1) +6*a(n-2) -6*a(n-3) -a(n-4) +a(n-5). G.f.: x*(-3-10*x-3*x^2+10*x^3+4*x^4)/((x-1) * (x^2-2*x-1) * (x^2+2*x-1)). G.f.: ( 8 + (-5-2*x)/(x^2+2*x-1) + (12+29*x)/(x^2-2*x-1) + 1/(x-1) )/2. (End) EXAMPLE 3*(3+1)/2+30 = 6^2. 13*(13+1)/2+30 = 11^2. 34*(34+1)/2+30 = 25^2. 84*(84+1)/2+30 = 60^2. MATHEMATICA Position[Accumulate[Range[8*10^6]], _?(IntegerQ[Sqrt[#+30]]&)]//Flatten (* Harvey P. Dale, May 30 2016 *) Join[{3, 13}, Select[Range[0, 10^5], ( Ceiling[Sqrt[#*(# + 1)/2]] )^2 - #*(# + 1)/2 == 30 &]] (* G. C. Greubel, Sep 03 2016 *) PROG (PARI) {for (n=0, 10^9, if ( issquare(n*(n+1)\2 + 30), print1(n, ", ") ) ); } CROSSREFS Cf. A000217, A000290, A006451. Sequence in context: A026084 A211801 A016061 * A281868 A137976 A095661 Adjacent sequences:  A154151 A154152 A154153 * A154155 A154156 A154157 KEYWORD nonn,less AUTHOR R. J. Mathar, Oct 18 2009 STATUS approved

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Last modified September 21 01:52 EDT 2020. Contains 337266 sequences. (Running on oeis4.)