A154148 - OEIS

%I #19 Jan 02 2023 12:30:47

%S 5,7,40,50,237,295,1384,1722,8069,10039,47032,58514,274125,341047,

%T 1597720,1987770,9312197,11585575,54275464,67525682

%N Numbers k such that 21 plus the k-th triangular number is a perfect square.

%H F. T. Adams-Watters, <a href="http://list.seqfan.eu/oldermail/seqfan/2009-October/002504.html">SeqFan Discussion</a>, Oct 2009

%F {k: 21+k*(k+1)/2 in A000290}

%F Conjectures: (Start)

%F a(n)= +a(n-1) +6*a(n-2) -6*a(n-3) -a(n-4) +a(n-5).

%F G.f.: x*(-5-2*x-3*x^2+2*x^3+6*x^4)/((x-1) * (x^2-2*x-1) * (x^2+2*x-1)).

%F G.f.: ( 12 + (12+25*x)/(x^2-2*x-1) + 1/(x-1) + (-1-10*x)/(x^2+2*x-1) )/2. (End)

%e 5, 7, 40, and 50 are terms:

%e    5* (5+1)/2 + 21 =  6^2,

%e    7* (7+1)/2 + 21 =  7^2,

%e   40*(40+1)/2 + 21 = 29^2,

%e   50*(50+1)/2 + 21 = 36^2.

%t Select[Range[68*10^6],IntegerQ[Sqrt[21+(#(#+1))/2]]&] (* _Harvey P. Dale_, Mar 07 2017 *)

%o (PARI) {for (n=0, 10^9, if ( issquare(n*(n+1)\2 + 21), print1(n, ", ") ) );}

%Y Cf. A000217, A000290, A006451.

%K nonn,less

%O 1,1

%A _R. J. Mathar_, Oct 18 2009