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A154154
Numbers k such that 30 plus the k-th triangular number is a perfect square.
3
3, 13, 34, 84, 203, 493, 1186, 2876, 6915, 16765, 40306, 97716, 234923, 569533, 1369234, 3319484, 7980483, 19347373, 46513666
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, 13, 34, and 84 are terms:
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
KEYWORD
nonn,less
AUTHOR
R. J. Mathar, Oct 18 2009
STATUS
approved