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A154150 Numbers k such that 24 plus the k-th triangular number is a perfect square.

%I A154150 #19 Dec 23 2024 14:53:42
%S A154150 1,15,24,94,145,551,848,3214,4945,18735,28824,109198,168001,636455,
%T A154150 979184,3709534,5707105,21620751,33263448,126014974
%N A154150 Numbers k such that 24 plus the k-th triangular number is a perfect square.
%H A154150 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 A154150 {k: 24+k*(k+1)/2 in A000290}.
%F A154150 Conjectures: (Start)
%F A154150 a(n) = +a(n-1) +6*a(n-2) -6*a(n-3) -a(n-4) +a(n-5).
%F A154150 G.f.: x*(-1-14*x-3*x^2+14*x^3+2*x^4)/((x-1) * (x^2-2*x-1) * (x^2+2*x-1)).
%F A154150 G.f.: ( 4 + 1/(x-1) + (10+27*x)/(x^2-2*x-1) + (-7+4*x)/(x^2+2*x-1) )/2. (End)
%e A154150 1, 5, 24, and 94 are terms:
%e A154150    1* (1+1)/2 + 24 =  5^2,
%e A154150   15*(15+1)/2 + 24 = 12^2,
%e A154150   24*(24+1)/2 + 24 = 18^2,
%e A154150   94*(94+1)/2 + 24 = 67^2.
%t A154150 Select[Range[12602*10^4],IntegerQ[Sqrt[24+(#(#+1))/2]]&] (* _Harvey P. Dale_, Jul 07 2019 *)
%o A154150 (PARI) {for (n=0, 10^9, if ( issquare(n*(n+1)\2 + 24), print1(n, ", ") ) );}
%Y A154150 Cf. A000217, A000290, A006451.
%K A154150 nonn,less
%O A154150 1,2
%A A154150 _R. J. Mathar_, Oct 18 2009