A154150 - OEIS

%I #18 Jan 02 2023 12:30:47

%S 1,15,24,94,145,551,848,3214,4945,18735,28824,109198,168001,636455,

%T 979184,3709534,5707105,21620751,33263448,126014974

%N Numbers k such that 24 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: 24+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*(-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 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 1, 5, 24, and 94 are terms:

%e    1* (1+1)/2 + 24 =  5^2,

%e   15*(15+1)/2 + 24 = 12^2,

%e   24*(24+1)/2 + 24 = 18^2,

%e   94*(94+1)/2 + 24 = 67^2.

%t Select[Range[12602*10^4],IntegerQ[Sqrt[24+(#(#+1))/2]]&] (* _Harvey P. Dale_, Jul 07 2019 *)

%o (PARI) {for (n=0, 10^9, if ( issquare(n*(n+1)\2 + 24), print1(n, ", ") ) );}

%Y Cf. A000217, A000290, A006451.

%K nonn,less

%O 1,2

%A _R. J. Mathar_, Oct 18 2009