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A154148 Numbers k such that 21 plus the k-th triangular number is a perfect square.

%I A154148 #20 Dec 23 2024 14:53:42
%S A154148 5,7,40,50,237,295,1384,1722,8069,10039,47032,58514,274125,341047,
%T A154148 1597720,1987770,9312197,11585575,54275464,67525682
%N A154148 Numbers k such that 21 plus the k-th triangular number is a perfect square.
%H A154148 F. T. Adams-Watters, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2009-October/002504.html">SeqFan Discussion</a>, Oct 2009
%F A154148 {k: 21+k*(k+1)/2 in A000290}
%F A154148 Conjectures: (Start)
%F A154148 a(n)= +a(n-1) +6*a(n-2) -6*a(n-3) -a(n-4) +a(n-5).
%F A154148 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 A154148 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 A154148 5, 7, 40, and 50 are terms:
%e A154148    5* (5+1)/2 + 21 =  6^2,
%e A154148    7* (7+1)/2 + 21 =  7^2,
%e A154148   40*(40+1)/2 + 21 = 29^2,
%e A154148   50*(50+1)/2 + 21 = 36^2.
%t A154148 Select[Range[68*10^6],IntegerQ[Sqrt[21+(#(#+1))/2]]&] (* _Harvey P. Dale_, Mar 07 2017 *)
%o A154148 (PARI) {for (n=0, 10^9, if ( issquare(n*(n+1)\2 + 21), print1(n, ", ") ) );}
%Y A154148 Cf. A000217, A000290, A006451.
%K A154148 nonn,less
%O A154148 1,1
%A A154148 _R. J. Mathar_, Oct 18 2009